Julien Grivaux

Rodrigue Desmorat

M Nicolas Dutertre

M Me

Irina Kogan

M Nicolas Auffray

M Jacky Cresson

M Julien Grivaux

M Rodrigue

M Boris Kolev

M Fabien Priziac

Any, Charo, Ipe Aziz David

Merci À Tony

Joe Marc

Sarah Merci

Yorgo

Clarita Christ

Yara

Peter

Liam Lucas

P Azzi

B Kolev

Geometry of isotropy classes for representation of groups with applications in Mechanics of Materials co-dirigée par

Keywords: Compact group representation, isotropy classes, constitutive law, polynomial optimization, continuum mechanics Représentation de groupe compact, classes d'isotropie, loi de comportement, optimisation polynomiale, mécanique des milieux continus 2020 Mathematics Subject Classification. 20G20, 57S15, 20G05, 14R20, 14L30, 15A72 Representations of compact Lie groups, Complexification of a compact Lie group, Isotropy stratification, Algebraicity of isotropy strata, Rationality problems

tout au long de ma vie.

Je tiens à exprimer toute ma reconnaissance à Marc Olive avec lequel j'ai eu plein de discussions enrichissantes durant mes premières années de thèse. Je te remercie pour toutes tes explications et tes conseils constructifs qui m'ont beaucoup aidé à avancer dans la thèse. Je remercie aussi Ahmed Benallal qui a initié le financement de ce projet. Je voudrais remercier également tous les mathématiciens et les mécaniciens que j'ai eus l'occasion de rencontrer durant cette thèse surtout dans le GDR-GDM. Je remercie les organisateurs Aziz Hamdouni et Vladimir Salnikov de m'avoir accueillie au sein du GDR où j'ai appris plein de choses et j'ai découvert le monde de la recherche des deux points de vue math et méca dans une ambiance bienveillante et amicale.

Mes quatre ans de doctorat furent l'occasion pour moi de rencontrer des collègues qui ont devenu ma deuxième famille. Je remercie tous les doctorants du LMPS et de l'IMJ-PRG pour la super ambiance et les nombreux bons moments passés ensemble.

Je commence par les membres du LMPS où j'ai commencé ma thèse. Je remercie en premier Je tiens à remercier mes amis du Liban qui ont été toujours à mes côtés malgré la distance.

In a second time, an effective method for determining the isotropy classes for reducible representations is proposed. This method is based on a binary operation between conjugacy classes of closed subgroups of a compact group, which has been introduced to obtain the isotropy classes of a direct sum of representations, if the isotropy classes are known for the irreducible factors. In the specific case of the three dimensional orthogonal group O (3), clips between some types of subgroups of O(3) have already been calculated. However, until now, clips between type II and type III subgroups were missing. Those are encountered in 3D coupled constitutive laws.

In this thesis, we complete the clips tables by computing the missing clips. As an application, we obtain 25 isotropy classes for the standard O(3)-representation on the full 3D Piezoelectric law, which involves the three elasticity, piezoelectricity and permittivity constitutive tensors.

Another problem that has been encountered many times in mechanics is solved in this work using algebraic methods: given an experimental material with no specific isotropy, it is interesting to determine the closest isotropy of a contitutive tensor for this material. To solve such a problem, we used methods of polynomial optimization taking advantage of the characterization of the isotropy classes by polynomial covariants. Two polynomial optimization methods are presented to solve this question and applied to some constitutive laws in mechanics such as elasticity and piezoelectricity laws: one makes use of some effective algorithm based on real algebraic geometry It straddles two research laboratories: IMJ-PRG (Institut de Mathématiques de Jussieu -Paris Rive Gauche ) and LMPS (Laboratoire de Mécanique Paris Saclay ). The main purpose of this thesis is to solve mathematical conjectures and mechanical problems arising from Continuum Mechanics using, in particular, real and complex algebraic geometry.

Mechanical motivation

In physics, the behavior of a material is described using constitutive laws and such laws are modeled using tensors. The tensor formulation of linear constitutive laws plays an important role in various physical theories, ranging from electromagnetism to continuum mechanics (see for instance [10]):

In the framework of linear elasticity, the stress tensor 𝜎 (of order 2) and the strain tensor 𝜀 (of order 2) of a material are related by the generalized Hooke's law 𝜎 = E : 𝜀. The fourth-order tensor E is called the elasticity tensor.

In the framework of piezoelectricity, the electric displacement tensor 𝑑 𝑑 𝑑 (of order 1) and the stress tensor 𝜎 (of order 2) are related by the tensorial constitutive equation 𝑑 𝑑 𝑑 = P : 𝜎, where the piezoelectric tensor P is a third-order tensor.

In these tensorial formulations, the material behavior is described by R-valued tensors (E and P). However, such a correspondence is not univocal: it depends on the orientation of the material in space. From a mathematical point of view, this corresponds to the action of the orthogonal group O(3) on the tensor space under study. Thus, in order to recognize a material, it is necessary to identify the orbit of the associated set of constitutive tensors. Besides, it is also important to take into account the symmetries of the material [25]. Appealing to Curie principle: -"the symmetries of the causes are to be found in the effects" -, a symmetry of a constitutive tensor is often expected in mechanical applications, by observing the micro-structure of a material. For instance, the elasticity tensor of a single crystal alloy with cubic crystal network is expected to be cubic (conjugacy class [O]), the piezoelectric tensor of an aluminum nitrite (AlN) alloyed with rocksalt transition metal nitrites is expected to become cubic (conjugacy class [O -]) for a high chromium concentration [74].

The questions related to material symmetries can be formulated in the language of the representation theory of groups ( [13,[START_REF] Sternberg | Group theory and physics[END_REF]44,14]). We can classify the constitutive tensors according to their invariance under some O(3)-subgroup, and gather them in sets called isotropy strata.

In mechanics, since the considered groups are compact Lie groups, the space of the considered tensors decomposes into a finite disjoint union of isotropy strata. Each of these strata thus corresponds to a given symmetry, indexed by a conjugacy class [𝐻] of a closed subgroup 𝐻 of 𝐺.

This thesis is devoted to problems posed by solid mechanics and formulated in the geometrical framework of isotropy strata of a compact Lie group representation.

Problematics

Geometry of isotropy strata for compact Lie group representations

The geometrical structure of isotropy strata for a group representation 𝐺 on a real vector space 𝑉 can be deduced from the geometry of the orbit space 𝑉 /𝐺. The initial observation is that the space of orbits 𝑉 /𝐺 can be described directly using the invariant algebra R[𝑉 ] 𝐺 . In fact, if 𝐺 is a compact Lie group, the algebra of 𝐺-invariant polynomials is known to separate the orbits [1, Appendix C]. Consequently, if {𝐽 1 , 𝐽 2 , . . . , 𝐽 𝑁 } is a set of polynomial invariants which separate the 𝐺-orbits, then the map 𝐽 : 𝑣 ↦ → ( 𝐽 1 (𝑣), 𝐽 2 (𝑣), . . . , 𝐽 𝑁 (𝑣) )

induces an algebraic homeomorphism J between the orbit space 𝑉 /𝐺 and 𝐽(𝑉 ) ⊂ R 𝑁 , which is a semialgebraic subset of R 𝑁 . This link between the invariant algebra and the orbit space is initially described by Abud-Sartori [1] and then clearly established by Procesi-Schwarz [START_REF] Procesi | Inequalities defining orbit spaces[END_REF].

More precisely, the semialgebraic description of 𝑉 /𝐺 is expressed using the positivity of a matrix with entries the differentials of the 𝐺-invariant functions. This matrix is called the Bézout matrix (see [START_REF] Procesi | Inequalities defining orbit spaces[END_REF]) or the Hermite matrix (see [23, section 1.4.1]). A rational parametrization for the orbit space using invariant functions has been studied by Sartori-Valente in [START_REF] Sartori | Rational parametrization of strata in orbit spaces of compact linear groups[END_REF]. A recent study on this link was then initiated by Auffray-Kolev-Petitot [2], restricting to the case of an irreducible representation of SO(3, R). In [2], Auffray-Kolev-Petitot reduced the study of orbit space, in a non trivial particular case, by considering the representation of the normalizer 𝑁 (𝐻), for 𝐻 an isotropy subgroup of 𝐺, on the subspace of fixed points 𝑉 𝐻 of 𝑉 .

Beyond the semialgebraic structure of the isotropy strata, it was observed, in some cases in mechanics of materials, that isotropy strata can be characterized explicitly by polynomial equations. For instance, in [38], most of the isotropy strata of the SO(2)-representation on the space of elasticity tensors are given by polynomial equations. Moreover, recently, the exploitation of covariants, geometrically richer than the invariants, allowed to obtain explicit polynomial equations of the eight strata of the SO(3)-representation on the space of harmonic tensors of order 4 (see [START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]). This characterization of the isotropy strata by polynomial equations in some specific cases, brings up questions about the algebraic structure of strata in more general contexts. An important result of this thesis is that, for any representation of compact Lie group on a real vector space 𝑉 , closed isotropy strata are real algebraic subsets of 𝑉 , being an irreducible component of the real locus of an algebraic complex variety (Theorem 1.1 of chapter 2).

Rationality of polynomial invariants for compact Lie group representations

It was observed in [2], on an example issued from mechanics, that polynomial invariants for the restricted representation of the normalizer of an isotropy subgroup 𝐻 of the group 𝐺 on the fixed point set 𝑉 𝐻 could be expressed rationally using polynomial invariants of the original representation of the group 𝐺 on 𝑉 . In other words, when 𝐻 is an isotropy subgroup of 𝐺, the field of 𝐺-invariants on 𝑉 is isomorphic to the field of 𝑁 (𝐻)-invariants on 𝑉 𝐻 . To illustrate this assertion, consider the example of the natural representation of the rotation group SO(3, R) on the vector space of traceless symmetric matrices with real coefficients Then, one can check that 𝜆 = 𝐼 3 𝐼 2 , where 𝐼 2 = tr 𝐴 2 and 𝐼 3 = tr 𝐴 3 are a generating set of the invariant algebra R[H 2 (R 3 )] SO (3,R) .

𝜌(𝑔)(𝐴) = 𝑔𝐴𝑔 -1 , 𝐴 ∈ H 2 (R 3 ), 𝑔 ∈ SO(3, R).
Similar results can be found in the literature but under different hypotheses. In fact, the isomorphism between the invariant field for the representation (𝑉, 𝐺) and the invariant field for the restricted representation (𝑉 𝐻 , 𝑁 (𝐻)) is a consequence of the slice method used for solving the rationality problem of the invariant field (see [29,22]). The rationality problem consists in studying the conditions on the representation (𝑉, 𝐺), under which the field of invariants 𝐾(𝑉 ) 𝐺 is purely transcendental over 𝐾. This is a very old problem: it was mentioned in 1911 in [16] (p.360), then, in 1913, Noether [START_REF] Noether | Rationale functionenkgrper[END_REF] posed the problem in the form of a conjecture. In 1915, Fisher [35] proved that the field of invariants 𝐾(𝑉 ) 𝐺 is purely transcendental over 𝐾 when 𝐺 is an abelian group. Also, Maeda proved the result for the alternating group 𝐴 5 [71]. However, it is not always true. For instance, 𝐾(𝑉 ) 𝐺 is not purely transcendental if 𝐺 is a finite group [START_REF] Saltman | Noether's problem over an algebraically closed field[END_REF].

Moreover, counter-examples to Noether's conjecture were given in the case of non-algebraically closed fields [76] (see also [START_REF] Swan | Invariant rational functions and a problem of steenrod[END_REF]21,[START_REF] Saltman | Generic galois extensions and problems in field theory[END_REF]). The question is still open in many cases, for instance in the case where the group 𝐺 is connected.

The isomorphism between the two invariant fields 𝐾(𝑉 ) 𝐺 and 𝐾(𝑉 𝐻 ) 𝑁 (𝐻) was proved for several cases with the purpose of solving the rationality problem by the slice method. For instance, in order to prove the rationality of the invariant field for the group 𝑆𝐿 2 (C) acting on the binary forms of degree 2𝑛, Bogomolov-Katsylo in [56,57,11] and later Maeda in [72], proved that C(𝑉 ) 𝐺 is isomorphic to C(𝑉 𝐻 ) 𝑁 (𝐻) (for 𝐻 generic). Miyata [START_REF] Miyata | Invariants of certain groups i 1[END_REF] and later Vinberg [START_REF] Vinberg | Rationality of the field of invariants of a triangular group[END_REF] proved the rationality of the orbit space 𝑉 /𝐺 for solvable connected algebraic groups by applying the slice method looking for the slice 𝑊 such that 𝑁 (𝑊 ) is a solvable connected subgroup (see also [94, section 2.9]). The result was also proven in [41] for the spinor group and in [46,47] for finite groups of monomial automorphisms. The rationality problem has a positive answer in many cases where the isomorphism 𝐾(𝑉 ) 𝐺 ≃ 𝐾(𝑉 𝐻 ) 𝑁 (𝐻) is used to prove it (for more details see [22]).

In addition, Luna in [69] and then Luna and Richardson in [70, theorem 4.2] proved that, for a linear action of a reductive group 𝐺 on an affine variety 𝑉 (so in particular for a linear representation of 𝐺), the restriction map 𝐾[𝑉 𝐻 ] → 𝐾[𝑉 ] maps 𝐾[𝑉 ] 𝐺 isomorphically onto 𝐾[𝑉 𝐻 ] 𝑁 (𝐻)/𝐻 when 𝐾 is an algebraically closed field and 𝐻 is a generic isotropy (see also [START_REF] Schwarz | Lifting smooth homotopies of orbit spaces[END_REF]).

In this thesis, we prove that there is surjective morphism between the two invariant fields over R and for any isotropy subgroup 𝐻, not only the generic one, that is: for any representation of a compact Lie group 𝐺 on a real vector space 𝑉 and for any isotropy subgroup 𝐻, the restriction map R[𝑉 ] → R[𝑉 𝐻 ] induces a surjective morphism 𝑆 : R(𝑉 ) 𝐺 R(𝑉 𝐻 ) 𝑁 (𝐻) , between the invariant fields of (𝑉, 𝐺) that are rationally defined on 𝑉 𝐻 , and of those of (𝑉 𝐻 , 𝑁 (𝐻)).

The proof appeals to Popov and Vinberg orbits separating theorem ([94, lemma 2.1]) and to results on the geometry of orbits by Luna [69] and Richardson [70]. Popov and Vinberg's theorem states that a separating set is a generating set of C(𝑉 𝐻 ) 𝑁 (𝐻) . Note that the latter theorem is wrong in the real setting. For instance, in the case of the real representation of the trivial group 𝐺 = {𝑒} on the real space 𝑉 = R, the invariant field is given by R(𝑉 ) 𝐺 = R(𝑥).

The polynomial 𝑝(𝑥) = 𝑥 3 separates all the orbits but is not a generator of R(𝑉 ) 𝐺 . For this reason, a detailed theory on the complexification of a real linear representation of a compact Lie group is fully provided in chapter 2.

Calculating isotropy strata for a compact Lie group representation

Finding the isotropy strata has always been a difficult problem. Classifying materials according to their symmetries goes back at least to the work of Lord Kelvin ([107], [58]). From then, many researchers devoted a great effort to the problem, especially for SO (3) and O(3) tensorial representations (used to model constitutive laws in mechanics). For instance, concerning the fourth-order elasticity tensor [45,24,[START_REF] Schouten | Tensor analysis for physicists[END_REF][START_REF] Yong-Zhong | On the completeness of the crystallographic symmetries in the description of the symmetries of the elastic tensor[END_REF]17], it was only in 1996 that Forte and Vianello [37] proved that there exists exactly eight symmetry classes. Inspired by the observation of Khatkevich ([59]), Forte-Vianello clarified the mathematical problem about the symmetry classes of an elasticity tensor. They removed the link with crystallographic point groups, which was extremely confusing and lead to the false conjecture that there were ten symmetry classes.

Following Forte and Vianello, similar results were obtained for other constitutive tensor spaces ( [38,66]). For instance, sixteen isotropy classes were obtained for the piezoelectricity tensor (a third-order tensor) [START_REF] Nye | Physical Properties of Crystals[END_REF][START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF][START_REF] Weller | Etude des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF][START_REF] Newnham | Properties of Materials: Anisotropy, Symmetry, Structure[END_REF][START_REF] Zou | Symmetry types of the piezoelectric tensor and their identification[END_REF].

Besides these results on symmetry classes in continuum mechanics, the subject has also been active in the mathematical community. First, it was interesting to see whether there is a finite number of isotropy strata or not. When the group is not finite, the existence of a finite number of isotropy classes was first conjectured by Montgomery ( [32]). Later, Mostow extended the result for a compact Lie group acting (not necessarily in a differentiable way) on a compact manifold [START_REF] Mostow | On a conjecture of Montgomery[END_REF], using results of Floyd [36] (see also [73,13]). After that, isotropy classes of irreducible representations for SO(3)-linear representations were obtained by Michel [START_REF] Michel | Symmetry defects and broken symmetry. configurations hidden symmetry[END_REF]. Then, Ihrig and Golubitsky [53] extended the result of Michel to the case of O(3)-irreducible representations.

However, Forte-Vianello approach requires rather refined calculations and reasoning to establish the classification. This complexity makes it difficult to apply to more involved situations, such as constitutive tensors of order greater than 4, or coupled constitutive laws involving a family of tensors [54,33]. A systematic way to calculate such isotropy classes was proposed by Chossat and Guyard in [18] for a direct sum of two irreducible SO(3)-representations. To do so, they introduced a binary operation on conjugate SO(3)-subgroups that allows one to compute the set of isotropy classes of a direct sum 𝑉 = 𝑉 1 ⊕ 𝑉 2 of linear representations of a group 𝐺 (if the isotropy classes for each individual irreducible representation are known). Inspired by Chossat and Guyard, Olive and coauthors used this operation, called clips operation, to complete and systemize the initial work of Chossat et al ( [START_REF] Olive | Symmetry classes for even-order tensors[END_REF], [START_REF] Olive | Symmetry classes for odd-order tensors[END_REF], [START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF] and [START_REF] Olive | Symmetry classes in piezoelectricity from second-order symmetries[END_REF]). Clips form a binary operation on the set of conjugacy classes of subgroups of a compact group 𝐺. It comes down to calculate the intersections between two subgroups 𝐻 1 and 𝑔𝐻 2 𝑔 -1 for 𝑔 ∈ 𝐺. In the case of the group O(3), clips operation between most of O(3)-subgroups are computed in [18] and [START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF], except for the clips between type II and type III O(3)-subgroups. Those are nevertheless encountered in practice, such as in the coupled Piezoelectricity law, where the representations of O(3) on three different constitutive tensor spaces are involved, namely the fourth-order elasticity tensor, the third-order piezoelectricity tensor and the second-order permittivity tensor.

We complete in the present work the clips tables of O(3) by computing the missing type II/type III clips. As an application of this result, we obtain 25 isotropy strata for the coupled Piezoelectricity law. We also address the coordinate-free characterization of isotropy strata by polynomial covariants [38,7,[START_REF] Rychlewski | A qualitative approach to hooke's tensors. part I[END_REF]12,2,[START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF] Distance of a constitutive tensor to an isotropy stratum An important problem in mechanics is the problem of determining the distance of a constitutive tensor to an isotropy stratum [42,39]. A measured constitutive tensor exhibits, in general, no material symmetry. However, in engineering applications, a measured tensor may have an expected symmetry due to the microstructure of the corresponding material (appealing to Curie principle). Finding the closest symmetry stratum to an experimental tensor is important in practice, since it allows to reduce the number of material constants. A natural question can then be asked: "What is the closest tensor having a given symmetry class, to an experimental measured tensor?" In order to answer this question, one needs to calculate the distance 𝑑(T 0 , Σ) = min T∈Σ ‖T 0 -T‖ 2 between the measured tensor T 0 and the closed isotropy stratum Σ representing the expected symmetry of T 0 and, then, deduce the corresponding minimizer.

This notion of distance to symmetry classes was first introduced in mechanics by Gazis and coauthors, in 1963, using orthogonal projections of elasticity tensors on the isotropy stratum.

Following Gazis, many authors considered the calculation of the distance to find the closest tensor having a specific symmetry to an experimental one [40,[START_REF] Moakher | The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry[END_REF]15,61,28,27,26,60,[START_REF] Ostrosablin | Transversely isotropic tensor closest in euclidean norm to a given anisotropic elastic modulus tensor[END_REF]).

In [40], François used the pole figures to deduce the symmetry class of an elasticity tensor. He considered an iterative numerical method based on the simplex method for minimization problems, parameterized by Euler angles, which turns the problem into a non-polynomial one. Other authors followed François, however they did not formulate the distance to a symmetry stratum as a polynomial optimization problem, so that the problems they had to solve (numerically) had many local minima and several global minima, making the computations of the solutions difficult.

In order to solve the distance problem, we take advantage of the fact that the isotropy strata can be described by polynomial equations, and formulate the distance of a constitutive tensor to an isotropy stratum as a polynomial optimization problem, which consists in minimizing or maximizing a polynomial function under some polynomial constraints.

In chapter 5 of this manuscript, we investigate a polynomial optimization method designed by Lasserre and coworkers [65] to solve our problem, thanks to our formulation of the distance problem as a polynomial optimization problem. This method consists in building a hierarchy of relaxed semidefinite problems converging to the optimal solution under some hypothesis on the constraints set. This method is implemented in a software called GloptiPoly [48,49,65].

However, Lasserre's method deals with problems in terms of number of variables and degree of constraints which restricts its application to some specific constitutive laws. Using this method, we were able to find the closest cubic tensor to some experimental tensors for elasticity and piezoelectricity laws.

On the other hand, in order to minimize a polynomial function under polynomial constraints, one can also use analytical methods such as Euler-Lagrange method. It consists in formulating the constrained problem into an unconstrained one using Lagrange multipliers and then finding the critical points of the considered function by solving a system of polynomial equations [9,55,63].

However, this method requires that the set of constraints to be a submersion, which is far from being satisfied by all the situations with mechanical interest. In chapter 6, this method, combined with the computation of a Gröbner basis, allows to solve the distance to cubic elasticity. In addition, we determine the closest pair of cubic elasticity and plasticity tensor to an experimental one (i.e. in the case of the coupled elasto-plasticity law), a problem that could not be solved using Lasserre's algorithm since it involves more variables.

To conclude, this thesis emphasizes the fruitful links between Mechanics of Materials and Algebraic Geometry. While mechanics keeps inspiring conjectures in algebraic geometry, tools from algebraic geometry keep being applied to solve mechanics problems.

Outline

This manuscript is organized as follows. In chapter 1, we provide basic materials about group actions on a set focusing on compact Lie group representations. We recall some key properties of linear representations of compact Lie groups and we introduce some definitions and questions related to isotropy classes, stratification and geometry of orbit spaces. We provide proofs to some known theorems, such as the principal orbit theorem. In the end of this chapter, we present the theory of invariants and covariants, examples and counter-examples are exposed in order to illustrate some mechanical and mathematical notions all along the chapter.

In chapter 2, we study the geometry of isotropy strata for compact Lie group representations.

We answer a ten-year-old conjecture on the rationality of the invariants of compact Lie group representations, deduced from observations made in the case of SO(3)-representations on tensor spaces in continuum mechanics. Moreover, we prove a result about the algebraicity of the isotropy strata:

For any representation of a compact Lie group 𝐺 on a real vector space 𝑉 and for any isotropy group 𝐻, the polynomial invariants of the representation of the normalizer 𝑁 (𝐻) on the fixed point set 𝑉 𝐻 can be written rationally in terms of the polynomial invariants of the initial representation of 𝐺 on 𝑉 .

Under the same assumptions, closed isotropy strata are proved to be algebraic sets.

Complexification of compact Lie groups and real representations is needed. For this purpose, a detailed theory on complexification is provided in this chapter.

In chapter 3, we specify a number of general results from chapter 1 to the case of linear representations of the groups O(3) and SO(3). First, we calculate a detailed list of O(3)-subgroups with their corresponding generators (in Table 3.1). Second, we discuss the harmonic decomposition of tensor spaces, which consists in decomposing tensor representations into irreducible spaces (a model of which is given by so-called harmonic tensor spaces). On one hand, we give isotropy classes for some O(3)-irreducible representations and we give a polynomial characterization of isotropy classes in the case of SO(3)-representation on harmonic spaces of order 4 (theorem 3.2.7).

On the other hand, we introduce an effective method to find the isotropy classes for reducible representations. For this purpose, we use the clips operation on the isotropy classes and give the result of these clips operations on the closed subgroups of SO(3) as well as on the closed subgroups of O(3) which were not all calculated before. As a consequence, in this chapter, we provide a complete table of the clips operations between all types of O(3)-subgroups (Table 3.2 and Table 3. In the last two chapters, we solve the problem of finding the closest constitutive tensor having a specific symmetry to an experimental one by calculating the distance of the tensor to the chosen symmetry class. For that, we use polynomial optimization methods. In chapter 5, we apply Lasserre's algorithm for solving polynomial optimization problems to some numerical example in the case of elasticity and piezoelectricity laws. In chapter 6, we use Euler-Lagrange method to minimize our constrained problem by calculating the Euler-Lagrange equations. As an application, we apply this method in the case of elasticity and coupled elasto-plasticity laws.

Chapter 1

Geometry representation of compact

Lie groups

In this chapter, fundamental concepts in representation theory will be introduced to describe the orbit space of the action of a compact Lie group on a vector space.

{ 𝑥 2 + 𝑦 2 = 𝑟 2 } centered at the origin is a SO(2)-stable set of R 2 .
To an action of a group 𝐺 on a set 𝑉 is associated the group morphism

𝜙 : 𝐺 → Bij(𝑉 ) defined by, if 𝑔 ∈ 𝐺, for all 𝑣 𝑣 𝑣 ∈ 𝑉 , 𝜙(𝑔)(𝑣 𝑣 𝑣) = 𝑔 • 𝑣 𝑣 𝑣 (Bij(𝑉 ) denotes the group of bijections of 𝑉 ).
The kernel of an action is the kernel of the associated morphism 𝜙 consisting of the elements 𝑔 ∈ 𝐺 that fix all the elements of 𝑉 . It is defined by

𝐾 = {𝑔 ∈ 𝐺 | ∀𝑣 𝑣 𝑣 ∈ 𝑉, 𝑔 • 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣} .
𝐾 is a closed normal subgroup of 𝐺.

The orbit space

Given an action of a group 𝐺 on a set 𝑉 , an equivalence relation can be defined on 𝑉 . For 𝑣 𝑣 𝑣 1 , 𝑣 𝑣 𝑣 2 ∈ 𝑉 , we define "∼" as follows

𝑣 𝑣 𝑣 1 ∼ 𝑣 𝑣 𝑣 2 ⇐⇒ ∃𝑔 ∈ 𝐺; 𝑣 𝑣 𝑣 1 = 𝑔 • 𝑣 𝑣 𝑣 2 . (1.3)
For 𝑣 𝑣 𝑣 ∈ 𝑉 , the equivalence class of 𝑣 𝑣 𝑣 is called the 𝐺-orbit of 𝑣 𝑣 𝑣 and is denoted by 𝐺 • 𝑣 𝑣 𝑣:

𝐺 • 𝑣 𝑣 𝑣 := {𝑔 • 𝑣 𝑣 𝑣, 𝑔 ∈ 𝐺} . (1.4)
The 𝐺-orbits form a partition of 𝑉 and the space of all 𝐺-orbits of 𝑉 is the orbit space denoted by 𝑉 /𝐺.

𝐺 𝑔•𝑣 𝑣 𝑣 = 𝑔𝐺 𝑣 𝑣 𝑣 𝑔 -1 .
For 𝑣 𝑣 𝑣 ∈ 𝑉 , we consider the orbit map 𝜌 𝑣 𝑣 𝑣 defined as follows:

𝜌 𝑣 𝑣 𝑣 : 𝐺 → 𝐺 • 𝑣 𝑣 𝑣 𝑔 ↦ → 𝑔 • 𝑣 𝑣 𝑣. The map 𝜌 𝑣 𝑣 𝑣 is surjective by definition of 𝐺•𝑣 𝑣 𝑣 but not necessarily injective. Indeed, if 𝑔, ℎ ∈ 𝐺, 𝑔 • 𝑣 𝑣 𝑣 = ℎ • 𝑣 𝑣 𝑣 ⇐⇒ 𝑔 -1 ℎ ∈ 𝐺 𝑣 𝑣 𝑣 , that is ℎ ∈ 𝑔𝐺 𝑣 𝑣 𝑣
. This shows that 𝜌 𝑣 𝑣 𝑣 is injective if and only if 𝐺 𝑣 𝑣 𝑣 = {𝑒} that is when the action is free on 𝑣 𝑣 𝑣 (see definition 1.1.4). Consequently, there is a unique bijective mapping

𝐺/𝐺 𝑣 𝑣 𝑣 ∼ -→ 𝐺 • 𝑣 𝑣 𝑣.
(1.5)

Next, we define another equivalence relation on 𝑉 . For this we need the following definition.

Definition 1.1.9. If 𝜌 1 and 𝜌 2 are two actions of 𝐺 on two spaces 𝑉 and 𝑊 , then a mapping

𝜑 : 𝑉 → 𝑊 is 𝐺-equivariant if ∀𝑔 ∈ 𝐺, 𝜑 ∘ 𝜌 1 (𝑔) = 𝜌 2 (𝑔) ∘ 𝜑.
For 𝑣 𝑣 𝑣 1 , 𝑣 𝑣 𝑣 2 ∈ 𝑉 , we define "≈" as follows:

𝑣 𝑣 𝑣 1 ≈ 𝑣 𝑣 𝑣 2 ⇐⇒ there exists a 𝐺-equivariant bijection 𝜑 : 𝐺 • 𝑣 𝑣 𝑣 1 → 𝐺 • 𝑣 𝑣 𝑣 2 . (1.6)
≈ is an equivalence relation and we denote the equivalence classes, called the orbit types of 𝑉 , by:

𝑉 ≈ 𝑣 𝑣 𝑣 = {𝑤 𝑤 𝑤 ∈ 𝑉 | 𝑣 𝑣 𝑣 ≈ 𝑤 𝑤 𝑤} . (1.7)
In the same manner, a preorder can be defined on 𝑉 , 𝑣 𝑣 𝑣 . The converse can be deduced from the bijection in (1.5).

2. Suppose that 𝑣 𝑣 𝑣 1 ≾ 𝑣 𝑣 𝑣 2 then there exists a 𝐺-equivariant map 𝜑 from 𝐺•𝑣 𝑣 𝑣 1 to 𝐺•𝑣 𝑣 𝑣 2 . We have

𝜑(𝑣 𝑣 𝑣 1 ) ∈ 𝐺 • 𝑣 𝑣 𝑣 2 , so 𝜑(𝑣 𝑣 𝑣 1 ) = 𝑔 ′ • 𝑣 𝑣 𝑣 2 for some 𝑔 ′ ∈ 𝐺. Let ℎ ∈ 𝐺 𝑣 𝑣 𝑣 1 , then ℎ • 𝑣 𝑣 𝑣 1 = 𝑣 𝑣 𝑣 1 . Applying 𝜑, we get 𝜑(ℎ • 𝑣 𝑣 𝑣 1 ) = 𝜑(𝑣 𝑣 𝑣 1 ) = 𝑔 ′ • 𝑣 𝑣 𝑣 2 which implies ℎ𝑔 ′ • 𝑣 𝑣 𝑣 2 = 𝑔 ′ • 𝑣 𝑣 𝑣 2 (𝜑(ℎ • 𝑣 𝑣 𝑣 1 ) = ℎ • 𝜑(𝑣 𝑣 𝑣 1 )).
Hence ℎ ∈ 𝑔 ′ 𝐺 𝑣 𝑣 𝑣 2 𝑔 ′-1 . For the converse implication, assume that there exists 𝑔 ∈ 𝐺 such

that 𝐺 𝑣 𝑣 𝑣 1 ⊂ 𝑔𝐺 𝑣 𝑣 𝑣 2 𝑔 -1 . We define 𝜑 : 𝐺 → 𝐺 • 𝑣 𝑣 𝑣 2 by 𝜑(𝑔 ′ ) = 𝑔 ′ 𝑔 • 𝑣 𝑣 𝑣 2 if 𝑔 ′ ∈ 𝐺.
𝜑(𝑔 ′ 𝑔 1 ) = 𝜑(𝑔 ′ ).

Fixed point set and isotropy classes

For a subgroup 𝐻 of 𝐺, we gather all the elements 𝑣 𝑣 𝑣 ∈ 𝑉 fixed by the elements of 𝐻 in a set called 𝐻-fixed point set, denoted by 𝑉 𝐻 :

𝑉 𝐻 = {𝑣 𝑣 𝑣 ∈ 𝑉 | ∀ℎ ∈ 𝐻, ℎ • 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣} .
Let 𝐻 be a subgroup of 𝐺, the conjugacy class of 𝐻, denoted by [𝐻], is given by

[𝐻] = { 𝑔𝐻𝑔 -1 , 𝑔 ∈ 𝐺 } .
The isotropy class of an element 𝑣 𝑣 𝑣 ∈ 𝑉 is the conjugacy class of its isotropy group. In other terms, ,

𝑣 𝑣 𝑣

Σ [𝐻] := {𝑣 𝑣 𝑣 ∈ 𝑉 ; [𝐺 𝑣 𝑣 𝑣 ] = [𝐻]} .
By lemma 1.1.10, we can observe that Σ [𝐻] = 𝑉 ≈ 𝑣 𝑣 𝑣 (1.7) for the vectors 𝑣 𝑣 𝑣 ∈ 𝑉 such that 𝐺 𝑣 𝑣 𝑣 = 𝐻. In other words, the isotropy stratum Σ [𝐻] consists of the vectors 𝑣 𝑣 𝑣 of 𝑉 having isotropy group conjugated to 𝐻. Note that the isotropy strata are 𝐺-stable sets.

Linear representation of compact Lie groups

In this section, 𝐺 denotes a compact Lie group and 𝑉 a vector space. A representation of the group 𝐺 on 𝑉 is a continuous group morphism

𝜌 : 𝐺 → GL(𝑉 ) 𝑔 ↦ → 𝜌(𝑔)
where GL(𝑉 ) denotes the group of linear automorphisms on 𝑉 and for 𝑣 𝑣 𝑣 ∈ 𝑉 , 𝜌(𝑔)𝑣 𝑣 𝑣 = 𝑔 • 𝑣 𝑣 𝑣. A representation of a group 𝐺 on a vector space 𝑉 will be denoted by (𝑉, 𝐺) or (𝑉, 𝜌).

Reducibility

𝜌(𝑡)(𝑥, 𝑦) = (𝑥 + 𝑡𝑦, 𝑦), 𝑡 ∈ R, (𝑥, 𝑦) ∈ R 2 .
𝜌 is reducible since 𝐸 =< (1, 0) > is stable by 𝜌. However, 𝜌 is not decomposable. Indeed, suppose that there exists a subspace 𝐹 of R 2 stable by 𝜌 and such that 𝐸 ⊕ 𝐹 = R 2 . Let 𝑢 be a generating vector of 𝐹 (dim

𝐹 = 1) such that 𝑢 = (𝑢 1 , 1) with 𝑢 1 ∈ R (𝑢 2 ̸ = 0 since 𝑢 / ∈ 𝐸).
Since 𝐹 is stable by 𝜌 then, ∀𝑡 ∈ R, ∃𝜆(𝑡) ∈ R such that 𝜌(𝑡)(𝑢) = 𝜆(𝑡)𝑢. In particular, for all 𝑡 ∈ R, 𝜆(𝑡) = 1 and then 𝑢 is fixed by all 𝜌(𝑡), 𝑡 ∈ R, which leads to a contradiction. 𝜌 is reducible since the subspace 𝑈 generated by the identity matrix I 𝑛 is 𝐺-stable as well as its orthogonal 𝑈 ⊥ = {𝐴 ∈ 𝑉 ; < 𝐴, I 𝑛 >= 0} = {𝐴 ∈ 𝑉 ; tr(𝐴) = 0} (where < •, • > is defined by, for 𝐴, 𝐵 ∈ 𝑉 , < 𝐴, 𝐵 >= tr(𝐴𝐵 𝑇 )). Moreover, 𝜌 is decomposable since 𝑉 can be decomposed in 𝑉 = 𝑈 ⊕ 𝑈 ⊥ where the representation of 𝐺 on 𝑈 ⊥ is irreducible. Indeed, since every symmetric matrix is diagonalisable in an orthonormal basis then each 𝐺-orbit intersect the subspace 𝐷 of diagonal matrices. Since 𝐷 is stable under the action of the permutation group 𝒮 𝑛 then the representation of 𝐺 on 𝑈 ⊥ is determined by the representation of 𝒮 𝑛 on 𝐷 ∩ 𝑈 ⊥ which is irreducible.

])). Then 𝑈 ⊥ = {𝑣 𝑣 𝑣 ∈ 𝑉 ; ∀𝑢 ∈ 𝑈, < 𝑣 𝑣 𝑣, 𝑢 𝑢 𝑢 >= 0} is a 𝐺-stable subspace of 𝑉 such that 𝑉 = 𝑈 ⊕ 𝑈 ⊥ . Remark 1.2.

Isotropic stratification

As 𝐺 is a compact group, a partial order relation can be defined on the set of the conjugacy classes of closed subgroups of 𝐺: Theorem 1.2.9. The relation on the set of conjugacy classes of closed subgroups of a compact group 𝐺 defined by

[𝐻 1 ] ⪯ [𝐻 2 ] if 𝐻 1 is conjugate to a subgroup of 𝐻 2 (1.9)
is a partial order relation.

To prove the anti-symmetricity of the relation (1.9), we make use of the following result:

Lemma 1.2.10. [13, proposition 1.9] Let 𝐺 be a compact group and 𝐻 be a closed subgroup of

𝐺. If 𝐻 ⊂ 𝑔𝐻𝑔 -1 for 𝑔 ∈ 𝐺 then 𝐻 = 𝑔𝐻𝑔 -1 . Proof. Let 𝑔 ∈ 𝐺 such that 𝐻 ⊂ 𝑔𝐻𝑔 -1 then ∀𝑛 ≥ 1, 𝑔𝐻𝑔 -1 ⊂ 𝑔 𝑛 𝐻𝑔 -𝑛 . Let ℎ ∈ 𝐻 then there exists ℎ 𝑛 ∈ 𝐻 such that 𝑔ℎ𝑔 -1 = 𝑔 𝑛 ℎ 𝑛 𝑔 -𝑛 . Since 𝐺 is compact, for all 𝑛 ∈ N there exists 𝑘 ∈ N and 𝑔 ′ ∈ 𝐺 such that the subsequence (𝑔 𝑛 𝑘 ) 𝑛 𝑘 ∈N of (𝑔 𝑛 ) 𝑛∈N converges to 𝑔 ′ . Set 𝑚 𝑘 = 𝑛 𝑘+1 -𝑛 𝑘 , then we have 𝑔ℎ𝑔 -1 = 𝑔 𝑚 𝑘 ℎ 𝑚 𝑘 𝑔 -𝑚 𝑘 → ℎ 𝑚 𝑘 . Since 𝐺 is compact, there exists 𝑟 ∈ N and ℎ ′ ∈ 𝐻 such that the subsequence (ℎ 𝑚 𝑘𝑟 ) 𝑚 𝑘𝑟 ∈N of (ℎ 𝑚 𝑘 ) 𝑚 𝑘 ∈N converges to ℎ ′ . It follows that 𝑔𝐻𝑔 -1 ⊂ 𝐻
and hence the equality.

proof of theorem 1.2.9. The reflexivity and the transitivity of ⪯ are straightforward. As for the anti-symmetricity, we use lemma 1.2.10: let 𝐻 1 and 𝐻 2 be two closed subgroups of 𝐺 such

that [𝐻 1 ] ⪯ [𝐻 2 ] and [𝐻 2 ] ⪯ [𝐻 1 ], then there exist 𝑔 1 , 𝑔 2 ∈ 𝐺 such that 𝐻 1 ⊂ 𝑔 1 𝐻 2 𝑔 -1 1 and 𝐻 2 ⊂ 𝑔 2 𝐻 1 𝑔 -1 2 .
We have, on one hand,

𝐻 1 ⊂ 𝑔 1 𝐻 2 𝑔 -1 1 ⊂ 𝑔 1 𝑔 2 𝐻 1 (𝑔 1 𝑔 2 ) -1 =⇒ 𝐻 1 = (𝑔 1 𝑔 2 )𝐻 1 (𝑔 1 𝑔 2 ) -1 ,
and on the other hand

𝐻 2 ⊂ 𝑔 2 𝐻 1 𝑔 -1 2 =⇒ 𝑔 1 𝐻 2 𝑔 -1 1 ⊂ (𝑔 1 𝑔 2 )𝐻 1 (𝑔 1 𝑔 2 ) -1 = 𝐻 1 ⊂ 𝑔 1 𝐻 2 𝑔 -1 1 .
Hence, we deduce the result.

The notions of orbit space, isotropy groups and isotropy classes, defined in section 1.1, extends naturally to the linear representations of a compact group. For an action of a finite group on a finite set, there exists a finite number of 𝐺-orbits. An interesting question arises here about the number of isotropy classes for a finite dimensional group representation.

Theorem 1.2.11. Let 𝐺 be a compact Lie group acting on a vector space 𝑉 . Then there exists a finite number of isotropy classes.

The finiteness of isotropy classes for a continuous action of a compact Lie group on a compact manifold was initially conjectured by Montgomery and solved by Mostow in [START_REF] Mostow | On a conjecture of Montgomery[END_REF] using the result of Floyd [36]. However, it is much easier in the case of a differentiable action (see [START_REF] Tom Dieck | of De Gruyter Studies in Mathematics[END_REF]Theorem 5.11]). This implies the finiteness of isotropy classes for a linear representation of a compact

Lie group : we can see it by extending a representation on 𝑉 to an action on the projective compactification P(𝑉 ⊕ R), or on the sphere 𝑆(𝑉 ) if we take an invariant inner product.

For a compact Lie group representation (𝑉, 𝜌), the partition of 𝑉 into (nonempty) isotropy strata

𝑉 = Σ [𝐻 0 ] ∪ • • • ∪ Σ [𝐻𝑛]
is called its isotropy stratification or orbit type stratification with respect to the representation (𝑉, 𝜌).

Principal isotropy stratum

To understand the geometry of conjugacy classes, a very useful notion is the notion of slice.

𝑆 is transverse to the orbit 𝐺 • 𝑣 𝑣 𝑣 (i.e. ∀𝑝 ∈ 𝑆 ∩ 𝐺 • 𝑣 𝑣 𝑣, 𝑇 𝑝 𝑆 ⊕ 𝑇 𝑝 (𝐺 • 𝑣 𝑣 𝑣) = 𝑇 𝑝 𝑉 ).
𝑆 is stable under 𝐺 𝑣 𝑣 𝑣 .

If 𝑠 1 , 𝑠 on the space of symmetric 3 × 3 matrices. Then there are exactly three orbit types

[D 2 ] ⪯ [O(2)] ⪯ [SO(3)]
where [D 2 ] is the principal stratum.

Closed isotropy stratum

Let 𝐻 be subgroup of 𝐺. We define the fixed point set 𝑉 𝐻 , which is a vector subspace of 𝑉 , by

𝑉 𝐻 := {𝑣 𝑣 𝑣 ∈ 𝑉 | ℎ • 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 for all ℎ ∈ 𝐻} ,
and the isotropy stratum Σ [𝐻] by

Σ [𝐻] := {𝑣 𝑣 𝑣 ∈ 𝑉 ; [𝐺 𝑣 𝑣 𝑣 ] = [𝐻]} .
We define the open fixed locus However, it may happen that

𝑉 𝐻 1 = 𝑉 𝐻 2 for 𝐻 1 ̸ = 𝐻 2 .
As a consequence of the order relation defined on the conjugacy classes in 1.2.9, we define closed isotropy stratum with respect to an isotropy subgroup 𝐻 to be the set consisting of vectors having at least the symmetry of 𝐻. The closed isotropy stratum with respect to 𝐻 is denoted by Σ [𝐻] and given by

Σ [𝐻] = { 𝑣 𝑣 𝑣 ∈ 𝑉 ; 𝐺 𝑣 𝑣 𝑣 ⊃ 𝑔𝐻𝑔 -1 for some 𝑔 ∈ 𝐺 } = ⋃ [𝐻]⪯[𝐾] Σ [𝐾] .
The closed stratum is the orbit of 𝑉 𝐻 .

Remark 1.2.19.

The terminology we use above is not totally standard, but in literature the terminology varies according to the authors (for instance, what we call stratum is called orbit bundle in [START_REF] Tom Dieck | of De Gruyter Studies in Mathematics[END_REF]).

The fixed locus 𝑉 𝐻 is defined for arbitrary closed subgroups of 𝐺 (not only isotropy subgroups) but we will generally avoid to do so.

The terminology "closed stratum" and the corresponding notation will be justified by Corollary 1.2.22. The same holds for the open fixed locus.

The partial order relation on isotropy classes induces a (reverse) partial order relation on the strata: if 𝐻 1 , 𝐻 2 are two isotropy subgroups of 𝐺,

[𝐻 1 ] ⪯ [𝐻 2 ] ⇐⇒ Σ [𝐻 2 ] ⪯ Σ [𝐻 1 ] Proposition 1.2.

Action of the monodromy group

Let 𝐺 be a compact Lie group acting linearly on 𝑉 .

If 𝐻 is a subgroup of 𝐺, We define the normalizer 𝑁 (𝐻) of 𝐻 by

𝑁 (𝐻) = { 𝑔 ∈ 𝐺 | 𝑔𝐻𝑔 -1 = 𝐻 } . ( 1
𝑔 • 𝐾 = 𝑔𝐾𝑔 -1 , if 𝑔 ∈ 𝐺 and 𝐾 is a subgroup of 𝐺.
We have the following result (see [2], [43]) :

(𝑔 • 𝑃 )(𝑣 𝑣 𝑣) := 𝑃 (𝑔 -1 • 𝑣 𝑣 𝑣) if 𝑃 ∈ K[𝑉 ]
and 𝑣 𝑣 𝑣 ∈ 𝑉 . We define the set of all polynomial functions on 𝑉 that are invariant under the action of 𝐺, denoted by

K[𝑉 ] 𝐺 , K[𝑉 ] 𝐺 = {𝑃 ∈ K[𝑉 ] | ∀𝑔 ∈ 𝐺, 𝑔 • 𝑃 = 𝑃 } . K[𝑉 ]
𝐺 is a subalgebra over K of the algebra K[𝑉 ] called the invariant algebra.

Integrity basis

The following result is due to Hilbert ( [50], see also chapter 2 subsection 1.3.1).

R[R 3 ] SO(3) = R[𝐼], 𝐼(𝑣 𝑣 𝑣) := ‖𝑣 𝑣 𝑣‖ 2 .
2. The standard representation (SO(3), S 2 (R 3 )), where we have Using this fact, the orbit space 𝑉 /𝐺 can then be described as a semialgebraic subset of R 𝑁 .

R[S 2 (R 3 )] SO(3) = R[𝐼 1 , 𝐼
Indeed, if {𝑝 1 , . . . , 𝑝 𝑁 } denotes a generating set for R[𝑉 ] 𝐺 , then the mapping

𝑃 : 𝑣 ↦ → ( 𝑝 1 (𝑣), 𝑝 2 (𝑣), . . . , 𝑝 𝑁 (𝑣) )
induces an homeomorphism between 𝑉 /𝐺 and 𝑃 (𝑉 ) ⊂ R 𝑁 which is a semialgebraic subset of R 𝑁 (see [START_REF] Procesi | Inequalities defining orbit spaces[END_REF]).

In the same way as is defined the algebra of polynomial invariants 𝐾[𝑉 ] 𝐺 , one can define the invariant field 𝐾(𝑉 ) 𝐺 of rational invariants, which is a subfield of the field 𝐾(𝑉 ) of all rational functions on 𝑉 . Note that this field is always finitely generated since it is contained in the finitely generated field 𝐾(𝑉 ) (see for instance [20, theorem 11.23]). Algorithms for constructing a generating set of rational invariants for the invariant field 𝐾(𝑉 ) 𝐺 exist and can be used to express any rational invariant in terms of the generators (see [51,52]).

Covariant algebra

In [62], Kraft and Procesi have generalized the concept of invariants in the following way.

Definition 1.3.6. Given two representations V and W of a group 𝐺, we define Pol(V, W) to be the space of polynomial mappings p from V to W. Pol(V, W) is a vector space of 𝑉 on which 𝐺 acts linearly:

∀𝑣 𝑣 𝑣 ∈ V, ∀𝑔 ∈ 𝐺, 𝑔(p(𝑔 • 𝑣 𝑣 𝑣)) = 𝑔 • p(𝑣 𝑣 𝑣) A polynomial covariant of V of type W is a 𝐺-equivariant polynomial mapping p : V → W, which means that ∀𝑣 𝑣 𝑣 ∈ V, ∀𝑔 ∈ 𝐺, p(𝑔 • 𝑣 𝑣 𝑣) = 𝑔 • p(𝑣 𝑣 𝑣).
We denote by Pol(V, W) 𝐺 the set of polynomial covariant of V of type W.

The space Pol(V, W) 𝐺 can be identified with the space L(W * , K[V]) of all linear mapping from W * into K[V]. Indeed, this correspondence comes from the identification of Pol(V, W)

𝐺 with K[V] ⊗ W.
This algebraic structure of Pol(V, W) 𝐺 will be then extended to create the covariant algebra (see also [94, Page 184] for a more general and abstract definition of this concept).

Definition 1.3.7. Let V, W be finite-dimensional representations of a group 𝐺 and W * the dual vector space of W. The covariant algebra of V of type W, noted Cov(V, W), is defined as the invariant algebra

K[V ⊕ W * ] 𝐺 .
Note that the covariant algebra Cov(V, W) has a natural bi-graduation. It is the direct sum of the finite dimensional vector spaces Cov 𝑑,𝑘 (V, W) of bi-homogeneous polynomial 𝑝(𝑣 𝑣 𝑣, 𝜔):

of total degree 𝑑 in 𝑣 𝑣 𝑣 ∈ V, called the degree of the covariant, and, of total degree 𝑘 in 𝜔 ∈ W * , called the order of the covariant.

Furthermore, the subspace of covariants of order 0 is identical to the invariant algebra of V.

Remark 1.3.8. The vector space of polynomial covariants Pol(V, W) 𝐺 can thus be identified with

Cov 1 (V, W) = +∞ ⨁ 𝑘=0 Cov 𝑘,1 (V, W),
the vector space of first-order covariants.

Example 1.3.9. When 𝐺 = SO(3) and W is the Euclidean space R 3 , the covariant algebra of a finite-dimensional representation V is given by 3) .

Cov(V) := R[V ⊕ R 3 ] SO(
An element p ∈ Cov(V)
Cov(V) = R[p 1 , . . . , p 𝑠 ].
Chapter 2

Rationality of normal forms of isotropy strata This chapter is an ArXiv preprint in which we study the isotropy stratification of a linear representation 𝑉 of a compact Lie group 𝐺. We achieve two important results.

First, we prove that for each isotropy subgroup 𝐻, every rational invariant of the induced representation (𝑉 𝐻 , 𝑁 (𝐻)) can be obtained as the restriction of a global invariant of (𝑉, 𝐺), where 𝑁 (𝐻) is the normalizer of 𝐻 and 𝑉 𝐻 is the fixed point set of 𝐻. Second, we prove that the closed isotropy strata are real algebraic sets being an irreducible component of the real locus of an algebraic complex variety.

The proof appeals to theorems and results on the geometry of orbits that hold only for algebraically closed field. Since we work with real representations of real groups, a detailed theory on the complexification of a real linear representation of a compact Lie group is fully provided in this chapter. Finally, in the appendix B, we give a proof for the principal orbit theorem in the linear case using linear slices.

Introduction

It was observed in [2], on an example from solid mechanics, that the coefficients of the normal form for some orbit types (also called isotropy classes or symmetry classes in mechanics) of a tensorial representation (𝑉, SO(3)) could be expressed rationally using some polynomial invariants of (𝑉, SO(3)). Let us start by giving an example of what we mean by this assertion. Consider the natural representation of the rotation group SO(3, R) on the vector space H 2 (R 3 ) of traceless symmetric 3 × 3 matrices with real coefficients

𝜌(𝑔)𝐴 = 𝑔𝐴𝑔 -1 , 𝐴 ∈ H 2 (R 3 ), 𝑔 ∈ SO(3, R).
For this representation, there are exactly three orbit types corresponding to the symmetry groups D 2 (three distinct eigenvalues, generic case), O(2) (two distinct eigenvalues) and SO(3) (only one eigenvalue). A normal form for the second orbit type represented by the symmetry group O(2) is provided by the linear subspace

𝑉 O(2) = { 𝐴 ∈ H 2 (R 3 ); 𝑔𝐴𝑔 -1 = 𝐴, ∀𝑔 ∈ O(2) } = {diag(-𝜆, -𝜆, 2𝜆); 𝜆 ∈ R} .
Then, on the dense open subset 𝜆 ̸ = 0 of (H 2 (R 3 )) O (2) , one has 𝜆 = 𝐼 3 𝐼 2 , where 𝐼 2 = tr 𝐴 2 and 𝐼 3 = tr 𝐴 3 are a generating set of the invariant algebra

R[H 2 (R 3 )] SO(3,R) .
More generally, consider a linear representation 𝜌 : 𝐺 → GL(𝑉 ) of a compact Lie group 𝐺 on a real vector space 𝑉 . Given an orbit type represented by a symmetry group 𝐻 ⊂ 𝐺, each orbit of this type intersects the fix point set 𝑉 𝐻 , which may be considered as a normal form for these orbits. The normalizer 𝑁 (𝐻) of 𝐻 in 𝐺 stabilizes the linear subspace 𝑉 𝐻 and 𝜌 induces a faithful representation of 𝑁 (𝐻)/𝐻 on 𝑉 𝐻 . It was conjectured, after the observations in [2], that every polynomial invariant of this representation is indeed the restriction on 𝑉 𝐻 of a rational invariant of the initial representation 𝜌 of 𝐺. In a subsequent paper [33], it was moreover conjectured that the closed strata of any real linear representation 𝑉 of a compact Lie group were moreover real algebraic subset of 𝑉 . In this paper, we will prove the following theorem:

Theorem 1.1. Let 𝜌 : 𝐺 → GL(𝑉 ) be a linear representation of compact Lie group 𝐺 on a real vector space 𝑉 , let 𝐻 be an isotropy group of 𝜌, let 𝑁 (𝐻) be the normalizer of 𝐻 in 𝐺, and let 𝑉 𝐻 be the fixed locus of 𝐻.

(1) Every rational invariant of the induced representation (𝑉 𝐻 , 𝑁 (𝐻)) is the restriction of an invariant of (𝑉, 𝐺) which is rationally defined on 𝑉 𝐻 . In other words, the restriction map

R[𝑉 ] → R[𝑉 𝐻 ] induces a morphism 𝑆 : R(𝑉 ) 𝐺 R(𝑉 𝐻 ) 𝑁 (𝐻) ,
which is surjective when restricted to its domain of definition.

(2) The closed stratum Σ [𝐻] is a closed real algebraic subset of 𝑉 .

Related results can be found in the literature but under different hypotheses. In almost all papers, the problem is considered in the complex case and for generic orbit types. The most classical result fitting in this framework is Chevalley's restriction theorem that corresponds to a connected semisimple complex Lie group acting by the adjoint action on its Lie algebra g. More generally, Luna in [29] and then Luna and Richardson in [30, theorem 4.2] proved that, for a linear action of a reductive group 𝐺 on an affine variety 𝑉 (hence in particular for a vector space), the restriction map 𝐾[𝑉 ] → 𝐾[𝑉 𝐻 ] maps 𝐾[𝑉 ] 𝐺 isomorphically onto 𝐾[𝑉 𝐻 ] Γ when 𝐾 is an algebraically closed field, and 𝐻 is the generic isotropy, and Γ = 𝑁 (𝐻)/𝐻 (see also [29], [36]). Further results can be found in [15,21,31,34] where the same isomorphism is proved on the fields of invariants of (𝑉, 𝐺) and (𝑉 (𝐻), 𝑁 (𝐻)) (that is 𝐾(𝑉 ) 𝐺 ≃ 𝐾(𝑉 𝐻 ) Γ ) under the hypothesis that 𝐾 is an algebraically closed field. To illustrate our theorem we provide below some explicit examples.

Example 1.2. We start with the real version of Chevalley's restriction theorem. If 𝐺 is a connected compact Lie group, all maximal tori are conjugate (see [8, Chap. IV Thm 1.6]) and they are maximal abelian subgroups. Let 𝑟 denote the rank of 𝐺, which is the dimension of any maximal torus 𝑇 in 𝐺. Recall that an element 𝑥 of g is regular if the closure of the 1-parameter subgroup of 𝐺 generated by 𝑥 is a maximal torus in 𝐺. Regular points form a dense open subset g reg of g, and for any 𝑥 in g reg , 𝐺 𝑥 is the centralizer of the maximal torus of 𝐺 containing the 1-parameter subgroup generated by 𝑥, which is the torus itself. Hence, if 𝑇 is a maximal torus, if 𝑊 = 𝑁 (𝑇 )/𝑇 is the corresponding Weil group, and if t is the Lie algebra of 𝑇 , there is an isomorphism R[𝑉 ] 𝐺 ≃ R[t] 𝑊 (see [8, Chap. VI Prop. 2.1]). Our result implies a weak form of this statement, namely the same isomorphism at the level of fields of invariants.

Example 1.3. For the standard representation of the symmetric group S 3 on 𝑉 = R 3 , there are three isotropy classes [1], [S 2 ], [S 3 ]. The invariant algebra R[R 3 ] S 3 is generated by the three elementary symmetric functions 𝜎 1 , 𝜎 2 and 𝜎 3 . The fixed point set 𝑉 S 2 is the plane 𝑥 = 𝑦 and

𝑁 (S 2 ) = S 2 . Hence, R[𝑉 S 2 ] 𝑁 (S 2 ) = R[𝑥, 𝑦] and we have 𝑥 = 𝜎 1 𝜎 2 -9𝜎 3 2𝜎 2 1 -6𝜎 2 and 𝑧 = 𝜎 3 1 -4𝜎 1 𝜎 2 + 9𝜎 3 𝜎 2 1 -3𝜎 2 • Example 1.4.
Here, we provide an example where 𝑁 (𝐻)/𝐻 is not trivial. We consider the representation of the rotation group SO(3, R) on the vector space 𝑉 = H 4 (R 3 ) of harmonic symmetric tensors of order 4 (or homogeneous harmonic polynomials in three variables of degree four). It is known, see [14,16], that there are eight orbit types, and among them the symmetry class [D 2 ], where D 2 is the dihedral group of order 2. The invariant algebra R[H 4 (R 3 )] SO(3,R) is generated by 9 polynomials 𝐽 𝑘 which have been obtained in [3] (see also [2]). The normalizer 𝑁 (D 2 ) is the octahedral group O and Γ = 𝑁 (D 2 )/D 2 ≃ S 3 . The fixed point set 𝑉 D 2 is the three-dimensional vector space spanned by the harmonic polynomials

⎧ ⎪ ⎨ ⎪ ⎩ 𝑝 1 = -𝑧 4 + 6𝑦 2 𝑧 2 -𝑦 4 , 𝑝 2 = -𝑧 4 + 6𝑥 2 𝑧 2 -𝑥 4 , 𝑝 3 = -𝑦 4 + 6𝑥 2 𝑦 2 -𝑥 4 .
In this basis, the action of the monodromy group Γ ≃ S 3 is just the standard action of the permutation group S 3 on the triple (𝜆 1 , 𝜆 2 , 𝜆 3 ) and the invariant algebra R[𝑉 D 2 ] S 3 is generated by the elementary symmetric polynomials (𝜎 1 , 𝜎 2 , 𝜎 3 ) in the 𝜆 𝑖 . It was shown in [2] that 𝜎 1 , 𝜎 2 and 𝜎 3 can be expressed rationally in terms of the 𝐽 𝑘 by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎 1 = - 9 ( 3 𝐽 7 -3 𝐽 2 𝐽 5 + 3 𝐽 3 𝐽 4 -𝐽 2 2 𝐽 3 ) 2 ( 6 𝐽 6 -9 𝐽 2 𝐽 4 -20 𝐽 3 2 + 3 𝐽 2 3 ) , 𝜎 2 = 4 7 𝜎 1 2 - 1 14 𝐽 2 , 𝜎 3 = 1 24 𝐽 3 + 1 7 𝜎 1 3 - 1 56 𝜎 1 𝐽 2 .
Outline. The article is organized as follows. In section 2, we recall basic material on representations of compact Lie groups and invariant theory. In section 3, we recall an explicit linear model for the complexification of a compact Lie group and provide several useful results on the complexification of a real continuous representation of a compact Lie groups. Besides, we prove one key technical tool, namely that for compact group, normalizers commute with complexification. The section 4 is devoted to the proof of theorem 1.1. In addition, two appendices are provided. In Appendix A, we recall some results about Lie groups, real analytic functions, and totally real manifolds. Finally, in Appendix B, we provide for the reader's convenience a proof of the Principal Orbit Theorem in the linear case.

Preliminaries

2.1. Linear representations of compact Lie groups. Let 𝐺 be a compact (real) Lie group and let 𝜌 : 𝐺 → GL(𝑉 ) be a continuous linear representation of 𝐺 on a finite dimensional vector space 𝑉 . It is well-known that 𝜌 is real-analytic (see corollary A.2). In particular, we will be able to apply all the tools of differential topology concerning smooth action of Lie groups on manifolds. We write 𝑔 • 𝑣 := 𝜌(𝑔)𝑣 to lighten the notation.

For any vector 𝑣 in 𝑉 , we denote by 𝐺 • 𝑣 its orbit under 𝐺. The quotient set 𝑉 /𝐺 of 𝑉 under the action of 𝐺 is the orbit space of 𝑉 .

The isotropy subgroup (or symmetry group) of a vector 𝑣 in 𝑉 is defined by

𝐺 𝑣 := {𝑔 ∈ 𝐺; 𝑔 • 𝑣 = 𝑣} .
It is a closed subgroup of 𝐺, hence a Lie subgroup of 𝐺. The orbit 𝐺 • 𝑣 is a smooth compact submanifold of 𝑉 , which is diffeomorphic to the homogeneous space 𝐺/𝐺 𝑣 . Two vectors in the same orbit have conjugate symmetry groups. Indeed,

∀𝑣 ∈ 𝑉, ∀𝑔 ∈ 𝐺, 𝐺 𝑔•𝑣 = 𝑔 𝐺 𝑣 𝑔 -1 .
Of course, the converse is false but we say that 𝑣 1 and 𝑣 2 are in the same isotropy class (or have the same orbit type in the terminology of [1]) if their symmetry groups are conjugate in 𝐺, that is if there exists 𝑔 ∈ 𝐺 such that

𝐺 𝑣 2 = 𝑔 𝐺 𝑣 1 𝑔 -1 .
Given a closed subgroup 𝐻 of 𝐺, we say that the conjugacy class [𝐻] of 𝐻 in 𝐺 is an isotropy class (or an orbit type) if there exists a vector 𝑣 such that [𝐻] = [𝐺 𝑣 ].

The finiteness of isotropy classes for a continuous action of a compact Lie group on a compact manifold was initially conjectured by Montgomery and solved in [13,32] (in the case of a smooth action it is way more easy, see also [4, Chapter VII] and [38,Theorem 5.11]). This implies the finiteness of isotropy classes for a linear representation of a compact Lie group: we can see it by extending a representation on 𝑉 to an action on the projective compactification P(𝑉 ⊕ R), or on the sphere 𝑆(𝑉 ) if we take an invariant inner product.

Given a compact Lie group 𝐺, the inclusion relation on the set of closed subgroups induces a partial order on the set of their conjugacy classes [7,1], called by some authors containment relation [9]. It is defined as follows:

[𝐻 1 ] ⪯ [𝐻 2 ],
if 𝐻 1 is conjugate to a subgroup of 𝐻 2 .

To understand the geometry of conjugacy classes, a very useful notion is that of a slice. We recall here the general definition, but in our setting not all properties are necessary (see 2.2). Definition 2.1. For any 𝑣 in 𝑉 , a local slice of the 𝐺-action at 𝑣 is an embedded disc 𝑆 ⊂ 𝑉 passing through 𝑣 such that:

• 𝑆 is transverse to the orbit 𝐺 • 𝑣.

• 𝑆 is stable under 𝐺 𝑣 . [38]). • The fixed locus 𝑉 𝐻 is defined for arbitrary closed subgroups of 𝐺 (not only isotropy subgroups) but we will generally avoid to do so. • The terminology "closed stratum" and the corresponding notation will be justified by corollary 2.11. The same holds for the opened fixed locus. • The partial order relation on isotropy classes induces a (reverse) partial order relation on the strata.

[𝐻 1 ] ⪯ [𝐻 2 ] ⇐⇒ Σ [𝐻 2 ] ⪯ Σ [𝐻 1 ]
The normalizer of 𝐻 is

𝑁 (𝐻) := { 𝑔 ∈ 𝐺 | 𝑔𝐻𝑔 -1 = 𝐻 } .
It is the maximal subgroup of 𝐺 in which 𝐻 is a normal subgroup. We have the following result (see [2], [16]):

Lemma 2.9. For each closed subgroup 𝐻 of 𝐺, 𝑉 𝐻 is 𝑁 (𝐻)-stable. Moreover, if 𝐻 = 𝐺 𝑣 0 is the isotropy group of some point 𝑣 0 ∈ 𝑉 , then 𝑁 (𝐻) = { 𝑔 ∈ 𝐺; 𝑔 • 𝑉 𝐻 ⊂ 𝑉 𝐻 } .

Proof. Let 𝑣 in 𝑉 𝐻 . Then, for any 𝑔 in 𝑁 (𝐻) and any ℎ in 𝐻, we have

ℎ • (𝑔 • 𝑣) = 𝑔 • (𝑔 -1 ℎ𝑔) • 𝑣 = (𝑔ℎ ′ ) • 𝑣 = 𝑔 • 𝑣. If moreover 𝐻 = 𝐺 𝑣 0 ,
𝑉 = Σ [𝐻 0 ] ∪ • • • ∪ Σ [𝐻𝑛]
is called its isotropy stratification or orbit type stratification. It can be shown that it is a real stratification, and even a Whitney stratification (see [10, §2.7 and Theorem 2.7.4]).

Example 2.12. Consider, for instance, the representation of the rotation group 𝐺 = SO(3) on 𝑉 = 𝑆 2 R 3 . Then there are exactly three orbit types

[D 2 ] ⪯ [O(2)] ⪯ [SO(3)],
where [D 2 ] is the principal orbit type. In that case, 𝑉 D 2 , the subspace of diagonal matrices, intersects all the orbits but since 𝑁 (D 2 ) is the octahedral group O and the monodromy Γ D 2 is isomorphic to the symmetric group S 3 , the set 𝑉 D 2 is not (formally speaking) a slice, but it can be seen as a slice with finite monodromy.

2.2.

Invariant theory and orbits. In the material we will present, we will constantly deal with two types of groups: Compact Lie groups: they admit automatically a faithful representation [8, Theorem III.4.1], and thanks to Weyl's unitary trick, this representation can be chosen unitary.

Besides, every continuous finite-dimensional representation of 𝐺 is fully irreducible, i.e. splits as a direct sum of irreducible representations. Complex reductive groups: they are complex Lie groups admitting a faithful complex analytic representations and such that every finite-dimensional analytic representation splits as a direct sum of irreducible representations.

It turns out that both type of groups are closely linked through the process of complexification, which will be detailed in 3.1. More precisely, complex reductive groups are exactly complexifications of compact Lie groups [25,Theorem 4.31]. The idea behind this link is that Weyl's unitary trick allows to prove that a complex Lie group is reductive as soon as it contains a Zariski dense compact subgroup (see for instance [24,Lemma 6.2.7]). Lastly, if 𝐺 ⊂ GL 𝑛 (C) is reductive, then, 𝐺 is automatically an algebraic subgroup of GL(𝑛, C) [25,Theorem 5.11].

Let 𝐺 be a real (resp. complex) Lie group, and let 𝐾 = R (resp. 𝐾 = C). The linear action of 𝐺 on a 𝐾-vector space 𝑉 extends naturally to the polynomial algebra 𝐾[𝑉 ] via the formula

(𝑔 • 𝑝)(𝑣) := 𝑝(𝑔 -1 • 𝑣)
for every polynomial 𝑝 ∈ 𝐾[𝑉 ] and every vector 𝑣 ∈ 𝑉 . The set of all polynomials that are invariants under the action of 𝐺 is a subalgebra of 𝐾[𝑉 ] denoted by 𝐾[𝑉 ] 𝐺 and called the invariant algebra of 𝑉 .

The foundational result of invariant theory, due initially to Hilbert [20] in the case of the action of GL(𝑛, C) on Sym 𝑑 C 𝑛 , runs as follows.

Theorem 2.13. [19, Theorem X.5.6], [24,Theorem 6.3.1] Let 𝐺 be a compact (resp. complex reductive) Lie group, let 𝐾 = R (resp. 𝐾 = C), let 𝑉 be a finite dimensional 𝐾-vector space, and let 𝜌 : 𝐺 → GL(𝑉 ) be a continuous (resp. analytic) representation of 𝐺. Then, the invariant algebra 𝐾[𝑉 ] 𝐺 is finitely generated. This means that there exists a finite set of invariant polynomials 𝑝 1 , . . . , 𝑝 𝑁 such that

𝐾[𝑉 ] 𝐺 = 𝐾[𝑝 1 , . . . , 𝑝 𝑁 ].
Remark 2.14. Although this theorem is stated most of the time for complex reductive groups (or even reductive groups over an algebraically closed field of characteristic zero), the proof in the compact case works in the same way since it relies only on the Noetherianity of 𝐾[𝑉 ] and the existence of a Reynolds operator (which is simply in this case obtained by averaging on the group).

It is clear that any invariant is constant on 𝐺-orbits. The geometry of orbits can be understood via the invariants, but the situation is different for the real and the complex cases. 𝐾 = R, 𝐺 compact:

• The 𝐺-orbits are compact and smooth.

• The invariants separate the 𝐺-orbits. In other terms, given two vectors 𝑣 1 , 𝑣 2 ∈ 𝑉 belonging to different 𝐺-orbits, it is always possible to find a function

𝑝 ∈ R[𝑉 ] 𝐺 such that 𝑝(𝑣 1 ) ̸ = 𝑝(𝑣 2 ) (see [1, Appendix C]).
• The orbit space 𝑉 /𝐺 can be described as a semialgebraic subset of R 𝑁 . Indeed, if {𝑝 1 , . . . , 𝑝 𝑁 } denotes a generating set for R[𝑉 ] 𝐺 , then the mapping

𝑃 : 𝑣 ↦ → ( 𝑝 1 (𝑣), 𝑝 2 (𝑣), . . . , 𝑝 𝑁 (𝑣) )
induces an homeomorphism between 𝑉 /𝐺 and 𝑃 (𝑉 ) ⊂ R 𝑁 which is a semialgebraic subset of R 𝑁 . 𝐾 = C, 𝐺 reductive:

• The 𝐺-orbits are constructible1 , and their closure are the same for the usual or for the Zariski topology. • Two Zariski-closed 𝐺-stable sets of 𝑉 can be separated by invariants.

• Each 𝐺-orbit is adherent to a unique closed 𝐺-orbit.

• The complex scheme 𝑉 /𝐺 = spec C[𝑉 ] 𝐺 parameterizes closed 𝐺-orbits.

In the same way one defines the algebra of polynomial invariants 𝐾[𝑉 ] 𝐺 , one can define the invariant field 𝐾(𝑉 ) 𝐺 of rational invariants, which is a subfield of the field 𝐾(𝑉 ) of all rational functions on 𝑉 . Note that this field is always finitely generated since it is contained in the finitely generated field 𝐾(𝑉 ) (see for instance [23]).

Following Popov and Vinberg [34], we say that a rational invariant 𝑓 separates the orbits 𝒪 1 and 𝒪 2 if it is defined at points of both orbits and assumes different values at these points. A subset 𝐹 of 𝐾(𝑉 ) 𝐺 separates the orbits 𝒪 1 and 𝒪 2 if it contains an element that separates these orbits. Finally, we say that a subset 𝐹 of 𝐾(𝑉 ) 𝐺 separates orbits in general position if there exists a nonempty Zariski open subset 𝑈 ⊂ 𝑉 such that 𝐹 separates the orbits of any two inequivalent points of 𝑈 .

Theorem 2.15. [34, Lem. 2.1] Let 𝐺 be a reductive complex group and 𝜌 : 𝐺 → GL(𝑉 ) a complex representation of 𝐺. If a finite set 𝐹 ⊂ C(𝑉 ) 𝐺 separates orbits in general position, then it generates the field C(𝑉 ) 𝐺 . Remark 2.16.

• The converse of theorem 2.15 is also true (see [34,Theorem 2.3]) and is originally due to Rosenlicht [35]. • Theorem 2.15 is wrong in the real setting. Consider the real representation of the trivial group 𝐺 = {𝑒} on the real space 𝑉 = R. Then the invariant field is given by R(𝑉 ) 𝐺 = R(𝑥). The polynomial 𝑝(𝑥) = 𝑥 3 separates all the orbits but is not a generator of R(𝑉 ) 𝐺 .

2.3. The algebraic case. The geometry of the linear action of a compact Lie group (or more generally the action of a Lie group on a manifold) has been investigated in the framework of algebraic geometry. The theory is delicate, we will present a quick overview of the situation as well as results that we need. The setting is as follows: let 𝐺 be a complex reductive group acting algebraically on a complex affine algebraic variety (the theory works for any closed field of characteristic zero, but we will use it essentially in the complex case). The aim is to provide a local model for the action near an orbit of 𝐺. The most simple example happens when the action is free. If we pursue the analogy with differential geometry, we would expect a local trivialization near the orbit. However, the following example shows that it is not possible to expect such a result in the Zariski topology.

Example 2.17. Let us consider the group 𝜇 𝑛 of 𝑛-th roots of unity, acting naturally on C * . In this case, the space of orbits is also isomorphic by C * , the quotient map 𝜋 being given by 𝑧 ↦ → 𝑧 𝑛 . We see that it is impossible to trivialize 𝜋 in the Zariski topology on the base. Indeed, a Zariski open subset of C * is simply obtained by removing a finite number of points, and the projection remains nontrivial on any such open subset.

The problem comes from the fact that a smooth and surjective morphism between algebraic varieties does not always have a section in the Zariski topology. However, it has a section in the etale topology [18, 17.16.3 (ii)]. Concretely this means the following: if 𝜙 : 𝑋 → 𝑌 is smooth and surjective, then for any 𝑦 in 𝑌 there is a neighborhood 𝑈 𝑦 of 𝑦 and an etale morphism 𝜙 : 𝑉 → 𝑈 𝑦 such that the pull-back morphism ̃ 𝑓 : 𝑉 × 𝑈𝑦 𝑋 → 𝑉 has a section. For the interested reader, let us mention that the problem of trivializing principal 𝐺-bundles in the Zariski topology was studied in depth by Grothendieck in [17]: in fact he proved that for a given algebraic group 𝐺, then all etale locally trivial principal 𝐺-bundles are Zariski locally trivial if and only if 𝐺 is affine, connected, and without torsion [17,Theorem 3].

Let us provide a very interesting example, due to Richardson, that illustrates another difficulty related to the topology of the orbits.

Example 2.18. [28, Remark 4 ∘ page 98] Let us consider the natural action of SL(2, C) on the set 𝑉 3 of cubic binary forms, i.e., homogeneous complex polynomials of degree 3 in two complex variables. Now, P(𝑉 3 ) is isomorphic to S 3 P 1 , the isomorphism being induced by the map which assigns to a cubic binary form its (unordered) three roots. The action of SL(2, C) is the natural action of PGL(2, C) on triplets of points in P 1 after quotienting by ±1. Hence there are only 3 possible projective orbits:

[𝑤 3 ], [𝑧𝑤 2 ] and [𝑧(𝑤 -𝑧)𝑤], whose respective stabilizers in PGL(2, C) are PGL(2, C) [𝑤 3 ] = {( 𝑎 𝑏 0 1 ) , 𝑎 ∈ C * , 𝑏 ∈ C } , PGL(2, C) [𝑧𝑤 2 ] = {( 𝑎 0 0 1 ) , 𝑎 ∈ C * } PGL(2, C) [𝑧(1-𝑧)𝑤] = {( 1 0 0 1 ) , ( 0 1 1 0 ) , ( -1 1 0 1 ) , ( 0 1 - 1 1
) ,

( 1 0 1 -1 ) , ( 1 -1 1 0
)} .

We can now look at the pre-images of these stabilizers in SL(2, C). They act by characters on binary forms. More precisely, for 𝜀 in {-1, +1}, if we denote by 𝛾 ⋆ 𝑓 , the right action of

𝛾 ∈ SL(2, C) on 𝑓 ∈ 𝑉 3 , given by (𝛾 ⋆ 𝑓 )(𝜉) = 𝑓 (𝛾 • 𝜉), where 𝜉 = (𝑧, 𝑤), we get ( 𝛼 𝛽 0 1/𝛼 ) ⋆ 𝑤 3 = 𝑤 3 𝛼 3 , ( 𝛼 0 0 1/𝛼 ) ⋆ 𝑧𝑤 2 = 𝑧𝑤 2 𝛼 𝑖𝜀 ( 0 1 1 0 ) ⋆ 𝑧(𝑤 -𝑧)𝑤 = -𝑖𝜀𝑧(𝑤 -𝑧), 𝑖𝜀 ( -1 1 0 1 ) ⋆ 𝑧(𝑤 -𝑧)𝑤 = -𝑖𝜀𝑧(𝑤 -𝑧) 𝜀 ( 0 1 -1 1 ) ⋆ 𝑧(𝑤 -𝑧)𝑤 = -𝜀𝑧(𝑤 -𝑧)𝑤, 𝑖𝜀 ( 1 0 1 -1 ) ⋆ 𝑧(𝑤 -𝑧)𝑤 = 𝑖𝜀𝑧(𝑤 -𝑧)𝑤 𝜀 ( 1 -1 1 0 ) ⋆ 𝑧(𝑤 -𝑧)𝑤 = -𝜀𝑧(𝑤 -𝑧)𝑤.
Hence we have 4 orbit types:

• The orbit of 0 has stabilizer SL(2, C).

• The orbit of 𝑤 3 has stabilizer C ⋊ Z/3Z.

• The orbit of 𝑧𝑤 2 has trivial isotropy.

• The orbit of 𝑧(𝑤 -𝑧)𝑤 has isotropy {( 1 0 0 1

) ,

( 0 -1 1 -1 ) , ( -1 1 -1 0 )} ≃ Z/3Z.
Now, the set 𝑈 of binary forms which have three distinct roots is a Zariski open subset of 𝑉 3 defined by the non vanishing of the discriminant. It is dense, and all points in 𝑈 have isotropy Z/3Z. However, SL(2; C) 𝑧𝑤 2 is trivial. Hence the generic orbit has isotropy Z/3Z, but another orbit type has trivial stabilizer. This example shows that we cannot expect corollary 2.3 to be true in whole generality in the algebraic setting.

In the preceding example, the reason why this problem occurs is that the orbit of 𝑧𝑤 2 is not closed. In the case of a closed orbit, Luna's slice theorem gives a good description of the situation.

Theorem 2.19. [28] Let 𝐺 be a reductive group acting on an affine algebraic variety 𝑋. Let 𝑥 be a point of 𝑋 and assume that the orbit 𝐺 • 𝑥 is closed in 𝑋. Then there exists an etale slice at 𝑥, that is an affine subvariety 𝑉 of 𝑋 passing through 𝑥 satisfying the following properties:

• 𝑉 is 𝐺 𝑥 -stable • The natural 𝐺-morphism 𝜙 : 𝐺 × 𝐺𝑥 𝑉 → 𝑈 is etale, and its image 𝑈 is an affine neighborhood of the orbit 𝐺 • 𝑥.

Using this theorem, Luna was able to recover results that were holding in the case of compact Lie group actions, like finiteness of orbit types. The result that will be of crucial importance for us is the algebraic version of corollary 2.3.

Corollary 2.20. [28, Remark 4 ∘ pp. 98] Let 𝐺 be a reductive group acting on an affine algebraic variety 𝑋. Let 𝑥 be a point of 𝑋 and assume that the orbit 𝐺 • 𝑥 is closed in 𝑋. Then there exists a Zariski open neighborhood of 𝑥 such that for any 𝑦 in 𝑋, 𝐺 𝑦 is conjugate to a subgroup of 𝐺 𝑥 .

Complexification

3.1.

Complexification of a compact Lie group. The complexification of a real Lie group is formally defined as the solution of a universal problem, which always exists and is unique up to a complex analytic isomorphism ([6, Chapter 3]).

If 𝐺 is compact, it is possible to define 𝐺 C as follows (see [8,III.8]): let 𝒜 be the algebra of representative functions on 𝐺, that is functions that generate a finite-dimensional representation inside 𝒞 0 (𝐴, R). Tannaka-Krein duality [8,III.7] guarantees that 𝐺 identifies with real characters of 𝒜, that is every character of 𝒜 is of the form 𝑓 → 𝑓 (𝑔) for 𝑔 in 𝐺. Then 𝐺 C is defined as the complex characters of 𝒜 C . . In this way we see that

C[cos(𝜃), sin(𝜃)] ≃ C[𝑧, 1/𝑧],
since every Laurent polynomial is uniquely determined by its restriction on U(1). Hence, the complex characters of C[cos(𝜃), sin(𝜃)] are exactly the points of C * .

There is an explicit way to describe this complexification, which is pretty useful to understand more precisely the geometry of 𝐺 C . To achieve this, we use the fact that every compact Lie group admits a faithful representation [8, Thm. III.4.1], and thanks to Weyl's unitary trick, it admits a faithful unitary representation. Before going further, let us recall some elementary facts of linear algebra. Lemma 3.2.

(1) Polar decomposition: if P(𝑛) denotes the set of hermitian positive definite matrices, then the product map U(𝑛) × P(𝑛) This being done, the complexification of a compact Lie group can be described as follows.

∼ -→ GL(𝑛, C) is a diffeomorphism. ( 2
Proposition 3.3. [8, Prop. III.8.3] Let 𝐺 ⊂ U(𝑛) be a compact Lie group and let g be its Lie algebra. Then

𝐺 C = { 𝑔𝑒 𝑖𝑍 ; 𝑔 ∈ 𝐺 and 𝑍 ∈ g } In particular, 𝐺 C is diffeomorphic to 𝐺 × g.
Remark 3.4. This result might be surprising at first glance because it is not clear at all that the right hand side is a group. Let us briefly explain by hand why it is the case. Given 𝑔 1 𝑒 𝑖𝑍 1 and 𝑔 2 𝑒 𝑖𝑍 2 , where 𝑔 1 , 𝑔 2 ∈ 𝐺 and 𝑍 1 , 𝑍 2 ∈ g, we have

𝑒 𝑖𝑍 1 𝑔 2 = 𝑔 2 Ad(𝑔 -1 2 )𝑒 𝑖𝑍 1 = 𝑔 2 𝑒 𝑖Ad(𝑔 -1 2 )(𝑍 1 ) = 𝑔 2 𝑒 𝑖𝑍 3 , where 𝑍 3 ∈ g. We get thus 𝑔 1 𝑒 𝑖𝑍 1 𝑔 2 𝑒 𝑖𝑍 2 = 𝑔 1 𝑔 2 𝑒 𝑖𝑍 2 𝑒 𝑖𝑍 3 . Writing 𝑒 𝑖𝑍 2 𝑒 𝑖𝑍 3 = 𝑔𝑒 𝑖𝑍 ,
with 𝑔 ∈ U(𝑛) and 𝑍 ∈ u(𝑛), we have thanks to lemma 3.2 (2),

𝑒 2𝑖𝑍 = 𝜄(𝑒 𝑖𝑍 2 𝑒 𝑖𝑍 3 ) -1 𝑒 𝑖𝑍 2 𝑒 𝑖𝑍 3 = 𝑒 𝑖𝑍 3 𝑒 2𝑖𝑍 2 𝑒 𝑖𝑍 3 .

Consider now the real analytic function

𝜙 : g × g → u(𝑛), (𝑍 2 , 𝑍 3 ) ↦ → 1 2𝑖 log(𝑒 𝑖𝑍 3 𝑒 2𝑖𝑍 2 𝑒 𝑖𝑍 3 ).
By applying lemma A.1 two times, we get that 𝜙(𝑍 2 , 𝑍 3 ) ∈ g C for sufficiently small 𝑍 2 , 𝑍 3 and by corollary A.5, we conclude that for all 𝑍 2 , 𝑍 3 in g, 𝜙(𝑍 2 , 𝑍 3 ) ∈ g C ∩ u(𝑛) = g. Hence this proves that for all 𝑍 2 , 𝑍 3 in g, 𝑍 is in g. We can apply again the same trick: consider the real analytic map

Ψ : g × g → U(𝑛), (𝑍 2 , 𝑍 3 ) ↦ → 𝑒 𝑖𝑍 2 𝑒 𝑖𝑍 3 𝑒 -𝑖𝜙(𝑍 2 ,𝑍 3 ) .
For 𝑍 2 , 𝑍 3 close to 0, log Ψ takes values in g C ∩ u(𝑛) = g, so Ψ takes values in 𝐺. As 𝐺 is a closed real analytic submanifold of U(𝑛) by corollary A.2, we deduce using again corollary A.5 that for all 𝑍 2 , 𝑍 3 in g, Ψ(𝑍 2 , 𝑍 3 ) ∈ 𝐺. Since Ψ(𝑍 2 , 𝑍 3 ) = 𝑔, we deduce that 𝑔 belongs to 𝐺 and we are done.

Remark 3.5. The group 𝐺 is a closed subgroup of GL(𝑛; C), and it is straightforward to check that Lie(𝐺 C ) = g + 𝑖g = g C since u(𝑛) is totally real in gl(𝑛; C). Hence Lie(𝐺 C ) carries a natural complex structure for which the bracket is complex linear. It defines on 𝐺 C a complex Lie group structure for which the exponential map is a local biholomorphism around 0. Besides, 𝐺 is totally real in 𝐺 C .

We have an even stronger property :

Proposition 3.6. The complexified group 𝐺 C is an affine algebraic subvariety of GL(𝑛, C). Proposition 3.9. Let 𝐺 be a compact Lie group, 𝐺 C its complexification, and 𝜑 : 𝐺 → 𝐺 C the natural morphism. For any complex Lie group 𝐻 and any morphism 𝑓 : 𝐺 → 𝐻 of Lie groups, there exists a unique homomorphism 𝐹 : 𝐺 C → 𝐻 of complex Lie groups such that 𝑓 = 𝐹 ∘ 𝜑.

Proof
𝐺 𝑓 / / 𝜑 𝐻 𝐺 C 𝐹 > >
Proof. We use the notation of proposition 3.3. Let 𝜎 denote the differential of 𝑓 at the origin and let 𝜎 C : g C → h be its complexification. We define a map 𝐹 : 𝐺 C → 𝐻 by 𝐹 (𝑔𝑒 𝑖𝑍 ) = 𝑓 (𝑔)𝑒 𝑖𝜎(𝑍) .

First we claim that 𝐹 is holomorphic. By Lie's third theorem (see [37, §II.8, Thm 1]), the morphism 𝜎 C can be uniquely integrated to a local holomorphic group morphism ̃ 𝐹 : 𝐺 C 𝐻 defined in a neighborhood of the identity, whose differential at the identity element is 𝜎 C . Hence,  𝐹 |𝐺 integrates 𝜎, so ̃ 𝐹 |𝐺 = 𝑓 . Hence, for 𝑔 close to the identity and 𝑍 close to 0, we get

 𝐹 (𝑔𝑒 𝑖𝑍 ) = ̃ 𝐹 (𝑔) ̃ 𝐹 (𝑒 𝑖𝑍 ) = 𝑓 (𝑔)𝑒 𝜎 C (𝑖𝑍) = 𝑓 (𝑔)𝑒 𝑖𝜎(𝑍) = 𝐹 (𝑔𝑒 𝑖𝑍 ),
so 𝐹 = ̃ 𝐹 near the identity. It follows that 𝐹 is holomorphic near the identity. Since 𝐹 is real analytic, 𝐹 is holomorphic on (𝐺 C ) 𝑒 . Since for 𝑔 in 𝐺 and ℓ in 𝐺 C , 𝐹 (𝑔ℓ) = 𝑓 (𝑔)𝐹 (ℓ), it follows that 𝐹 is holomorphic on 𝐺 C . Lastly, 𝐹 |𝐺 = 𝑓 is a group morphism on 𝐺, so by corollary 3.7, applied two times, 𝐹 is a group morphism. □ 3.2. Complexification of a real representation. Let 𝜌 : 𝐺 → GL(𝑉 ) be a real continuous representation of a compact Lie group 𝐺. By proposition 3.9, there exists a unique analytic extension 𝜌 C : 𝐺 C → GL(𝑉 C ) of 𝜌. This complexified representation enjoys some very specific properties that we will list below.

Proposition 3.10. If we see 𝐺 C as an affine algebraic group (via proposition 3.6), then 𝜌 C is an algebraic map.

Proof. This is proved in [8,Proposition III.8.6]. □ Remark 3.11. The link between compacity and algebraicity is a beautiful result that relies on the Peter-Weyl theorem and on the finiteness of the algebra of representative functions. Outside of this context, such results become immediately wrong. For instance, C and C × are both complex affine algebraic groups, but exp : C → C × is a holomorphic Lie group morphism that is no longer algebraic. This comes from the fact that C is not reductive.

Proposition 3.12. Given (𝑉, 𝐺), there exists a 𝐺 C -stable and nonempty Zariski open subset 𝑈 of 𝑉 C such that all orbits of points in 𝑈 are closed.

Proof. Since 𝐺 is compact, we can endow 𝑉 with an inner product. Hence we have a chain of maps 𝐺 𝜌 -→ 𝑂(𝑛) ⊂ U(𝑛) ⊂ 𝐺𝐿(𝑛; C) Thanks to proposition 3.3, it is possible to check that 𝑂(𝑛) C = 𝑂(𝑛; C) so we get that 𝜌 C takes values in 𝑂(𝑛; C). Hence the action preserves a symmetric nondegenerate complex bilinear form, and the result follows from [27]. □ Lemma 3.13. For any 𝑣 in 𝑉 , (𝐺 C ) 𝑣 = (𝐺 𝑣 ) C .

Proof. By corollary 3.7, 𝐺 𝑣 is Zariski dense in (𝐺 𝑣 ) C so (𝐺 𝑣 ) C fixes 𝑣 for the complexified representation. This yields the inclusion (𝐺 𝑣 ) C ⊂ (𝐺 C ) 𝑣 .

For the converse implication, we argue as in the proof of proposition 3.12. Let us denote by 𝑑𝜌 the differential of the 𝐺-action at the origin. Let ℓ = 𝑔𝑒 𝑖𝑍 be an element of (𝐺 C ) 𝑣 . Then 𝜄(ℓ) belongs to (𝐺 C ) 𝑣 too, so 𝑒 2𝑖𝑍 = 𝜄(ℓ) -1 ℓ belongs to (𝐺 C ) 𝑣 . It means that exp(2𝑖 𝑑𝜌(𝑍)) • 𝑣 = 𝑣. Now 2𝑖 𝑑𝜌(𝑍) is a complex hermitian endomorphism of 𝑉 , so it is diagonalizable with real eigenvalues. It follows that 𝑑𝜌(𝑍)(ℓ) = 0, so 𝑍 belongs to Lie(𝐺 𝑣 ). Hence, 𝑒 𝑖𝑍 fixes 𝑣 so 𝑔 fixes 𝑣 too. It follows that 𝑔 belongs to 𝐺 𝑣 , which proves that ℓ belongs to (𝐺

𝑣 ) C . □ Proposition 3.14. For any 𝑣 in 𝑉 , 𝐺 C • 𝑣 is closed. Proof. It follows directly from [11, Lemma 2.2] since 𝐺 is compact. □ Remark 3.15.
For the reader's convenience, we can provide a sketch of the proof of the above proposition. It relies heavily on the beautiful theory of Kempf and Ness [22]. Let us fix an inner product on 𝑉 such that 𝐺 acts by orthogonal transformations, and extend it to an hermitian product on 𝑉 C . The moment map 𝜇 : 𝑉 C → g * is given by

𝜇(𝑣)(𝜉) = 1 2𝑖 ⟨𝜉 • 𝑣|𝑣⟩
This definition makes sense because 𝜉 belongs to u(𝑉 C ) so it is skew-hermitian. Then Kempf-Ness theorem states that an orbit 𝐺 C • 𝑣 is closed if and only 𝐺 C • 𝑣 ∩ 𝜇 -1 (0) ̸ = ∅ (see for instance [26,Theorem 4]). However, since 𝐺 acts by orthogonal transformations, the moment map 𝜇 vanishes on 𝑉 . The result follows. □

= 𝑈 ∩ (𝑉 C ) 𝐻 C , which is a nonempty Zariski open subset of (𝑉 C ) 𝐻 C . Then for 𝑤 in ̃ 𝑈 , (𝐺 C ) 𝑤 is conjugate to a subgroup of 𝐻 C so there exists 𝑔 in 𝐺 C such that 𝐻 C ⊂ (𝐺 C ) 𝑤 ⊂ 𝑔 -1 𝐻 C 𝑔.
We can also compare real and complex orbits. This result won't be strictly necessary in the sequel but it can help to understand the geometry of the situation. Proposition 3.18. For any vector 𝑣 in 𝑉 ,

(𝐺 C • 𝑣) ∩ 𝑉 = 𝐺 • 𝑣.
Proof. One inclusion is obvious. For the other inclusion, let 𝐻 be the isotropy of 𝑣. Thanks to lemma 3.13, the orbit

𝐺 C • 𝑣 is isomorphic to 𝐺 C /𝐻 C . Let 𝜄 denotes the Cartan involution on 𝐺 C . Assume that ℓ • 𝑣 is real for some element ℓ in 𝐺 C . Then 𝜄(ℓ) • 𝑣 = ℓ • 𝑣 = 𝑣 so 𝜄(ℓ) -1 ℓ belongs to 𝐻 C . If ℓ = 𝑔𝑒 𝑖𝑍 , then 𝑒 2𝑖𝑍 belongs to 𝐻 C . It implies that 𝑍 is in h, so 𝑒 𝑖𝑍 is in 𝐻 C . Hence ℓ • 𝑣 = 𝑔 • 𝑣. □ Remark 3.19.
The result [5, Proposition 2.3] predicts in our case that the intersection of a complex orbit with real points is a finite union of real orbits. Hence this result is weaker, but holds for more general groups.

Example 3.20. Let us give an example where this intersection is strictly bigger than the real orbit. For this we take again Example 2.18, but with the action of GL(2; C) instead of SL(2; C) acting on the vector space of degree 3 binary forms. Then the GL(2, R)-orbit of 𝑧(𝑤 -𝑧)𝑤 consists of all binary forms of degree 3 with real coefficients and 3 distinct roots. However, real binary forms that are in the GL(2, C)-orbit of 𝑧(𝑤 -𝑧)𝑤 consist of real binary forms with 3 distinct roots (not necessarily real). This locus is a union of two GL(2; R) orbits: the orbit of 𝑧(𝑤 -𝑧)𝑤 and the orbit of 𝑧(𝑧 2 + 𝑤 2 ). There is no contradiction with the result we proved, because GL(2, R) is not compact. As a matter of fact, GL(2, R) is the split real form of GL(2, C) and not the compact one, which is U(2).

Finally, we go back to the algebra of invariants.

Lemma 3.21. The three complex algebras

(R[𝑉 ] 𝐺 ) C , C[𝑉 C ] 𝐺 and C[𝑉 C ] 𝐺 C are naturally iso- morphic. Proof. We have R[𝑉 ] C ≃ C[𝑉 C ].
Lie(𝑁 (𝐻)) = 𝑁 (h) ∩ {𝑋 ∈ g, ∀𝛾 ∈ Γ, Ad(𝛾)(𝑋) -𝑋 ∈ h} .
Proof. Let 𝑋 ∈ Lie(𝑁 (𝐻)) then for all real 𝑡, 𝑒 𝑡𝑋 is in 𝑁 (𝐻). Let 𝑌 ∈ h, then 𝑒 𝑠𝑌 is in 𝐻 and therefore Ad(𝑒 𝑡𝑋 )(𝑒 𝑠𝑌 ) is in 𝐻. Taking the derivatives with respect to 𝑠 and then to 𝑡, we get Ad(𝑋)(𝑌 ) ∈ h which implies that 𝑋 ∈ 𝑁 (h). Now, let 𝛾 be in Γ. Then Ad(𝑒 𝑡𝑋 )(𝛾) belongs to the connected component 𝐻 𝑒 𝛾 of 𝐻 containing 𝛾. Consequently, 𝑒 𝑡𝑋 𝛾𝑒 -𝑡𝑋 𝛾 -1 ∈ 𝐻. Deriving at 𝑡 = 0 we get that 𝑋 -Ad(𝛾)(𝑋) is in h. Conversely, let 𝑋 ∈ 𝑁 (h) ∩ {𝑋 ∈ g, ∀𝛾 ∈ Γ, Ad(𝛾)(𝑋) -𝑋 ∈ h}, and let 𝑌 be in h. Then,

𝑒 𝑡𝑋 𝑒 𝑠𝑌 𝑒 -𝑡𝑋 = Ad(𝑒 𝑡𝑋 )(𝑒 𝑠𝑌 ) = 𝑒 𝑠 Ad(𝑒 𝑡𝑋 )(𝑌 ) = 𝑒 𝑠 exp(𝑡ad(𝑋))(𝑌 ) ∈ 𝐻.
It follows that 𝑒 𝑡𝑋 𝐻 𝑒 𝑒 -𝑡𝑋 ⊂ 𝐻 𝑒 . We must now deal with the other connected components. Let 𝛾 be in Γ. Then

𝑒 𝑡𝑋 𝛾𝑒 -𝑡𝑋 𝛾 -1 = 𝑒 𝑡𝑋 Ad(𝛾)(𝑒 -𝑡𝑋 ) = 𝑒 𝑡𝑋 𝑒 -Ad(𝛾)(𝑡𝑋) = 𝑒 𝑡𝑋 𝑒 𝑡(𝑋-Ad(𝛾)(𝑋))-𝑡𝑋 .
Set 𝑌 = 𝑋 -Ad(𝛾)(𝑋), then, by assumption, 𝑌 ∈ h and 𝑡𝑌 -𝑡𝑋 = -Ad(𝛾)(𝑋) ∈ h. By applying Baker-Campbell-Hausdorff formula (lemma A.1) on 𝑒 𝑡𝑋 𝛾𝑒 -𝑡𝑋 𝛾 -1 = 𝑒 𝑡𝑋 𝑒 𝑡𝑌 -𝑡𝑋 , we get that 𝑒 𝑡𝑋 𝛾𝑒 -𝑡𝑋 𝛾 -1 = 𝑒 𝑡𝑍 where 𝑍 is an iteration of Lie brackets in terms of 𝑋 and 𝑌 hence 𝑍 ∈ h. Therefore, we have

𝑒 𝑡𝑋 (𝛾𝑒 𝑠𝑌 )𝑒 -𝑡𝑋 = (𝑒 𝑡𝑋 𝛾𝑒 -𝑡𝑋 )𝛾 -1 ⏟ ⏞ ∈𝐻 𝛾 (𝑒 𝑡𝑋 𝑒 𝑠𝑌 𝑒 -𝑡𝑋 ) ⏟ ⏞ ∈𝐻 𝑒 ∈ 𝐻.
Consequently, 𝑒 𝑡𝑋 normalizes 𝐻, which implies that 𝑋 belongs to Lie(𝑁 (𝐻)). □ Lemma 3.24. Let h be a Lie subalgebra of u(𝑛), 𝑍 ∈ h, 𝐵 ∈ 𝑁 (h) and 𝐴 ∈ u(𝑛).

If 𝑒 2𝑖𝐴 = 𝑒 𝑖𝐵 𝑒 𝑖𝑍 𝑒 𝑖𝐵 then 𝐴 -𝐵 ∈ h.
Proof. Consider the following real analytic function

𝑓 : u(𝑛) × 𝑁 (h) → u(𝑛), (𝑍, 𝐵) ↦ → 1 𝑖 ln(𝑒 𝑖𝐵 𝑒 𝑖𝑍 𝑒 𝑖𝐵 ) -2𝐵.
Note that 𝑓 is well-defined because 𝑒 𝑖𝐵 𝑒 𝑖𝑍 𝑒 𝑖𝐵 is an element of P(𝑛), thanks to lemma 3.2 (2). Indeed,

𝜄(𝑒 𝑖𝐵 𝑒 𝑖𝑍 𝑒 𝑖𝐵 ) = 𝑒 -𝑖𝐵 𝑒 -𝑖𝑍 𝑒 -𝑖𝐵 = (𝑒 𝑖𝐵 𝑒 𝑖𝑍 𝑒 𝑖𝐵 ) -1 .
Then, we can take its logarithm in u(𝑛) thanks to lemma 3.2 (3). Next, we apply Baker-Campbell-Hausdorff formula (lemma A.1) two times on 𝑒 𝑖𝐵 𝑒 𝑖𝑍 𝑒 𝑖𝐵 and we deduce that 𝑓 (𝐵, 𝑍) is in h for 𝐵 and 𝑍 sufficiently close to zero. Thanks to corollary A.5, this is valid everywhere since 𝑓 is real analytic. Hence 𝐴 -𝐵 belongs to h. □ Finally, we need the following classical result.

Lemma 3.25. Let 𝑢 be a diagonalizable endomorphism of a vector space 𝑉 with real eigenvalues.

If 𝑊 ⊂ 𝑉 is stable by exp(𝑢), then 𝑊 is stable by 𝑢.

Proof. Let 𝜆 1 , . . . , 𝜆 𝑘 be the distinct eigenvalues of 𝑢, and let 𝑃 be a polynomial such that for any 𝑖, 𝑃 (𝑒 𝜆 𝑖 ) = 𝜆 𝑖 . Then 𝑃 (exp(𝑢)) = 𝑢, since this equality can be checked on a basis of eigenvectors of 𝑢. The result follows directly. □ Proposition 3.26. We have

𝑁 𝐺 C (𝐻 C ) = (𝑁 𝐺 (𝐻)) C . Proof. If 𝑔 is in 𝑁 𝐺 (𝐻), then, 𝑔𝐻𝑔 -1 ⊂ 𝐻 ⊂ 𝐻 C . Consider the holomorphic application 𝐶 𝑔 : 𝐻 C → 𝐺 C /𝐻 C , ℎ ↦ → 𝑔ℎ𝑔 -1 .
Then, 𝐶 𝑔 vanishes on 𝐻, and thanks to corollary 3. C ). Let us prove the reverse inclusion. Let 𝑔 be in 𝑁 𝐺 C (𝐻 C ). By theorem 3.3, we can write 𝑔 = 𝑘𝑒 𝑖𝑋 with 𝑘 in 𝐺 and 𝑋 in g. We need to prove that 𝑘 (resp. 𝑋) belongs to 𝑁 (𝐻) (resp. Lie(𝑁 (𝐻)). We have 𝜄(𝑔) = 𝑘𝑒 -𝑖𝑋 . To prove that 𝑋 ∈ Lie(𝑁 (𝐻)), we use the description of Lie(𝑁 (𝐻)) given in lemma 3.23.

On one hand, thanks to lemma 3.2 (2), we have 𝑒 2𝑖𝑋 = 𝜄(𝑔) -1 𝑔. Besides, the normalizer of 𝐻 C is stable under the Cartan involution 𝜄, and hence

𝑒 2𝑖𝑋 normalizes 𝐻 C . This implies that Ad(𝑒 2𝑖𝑋 )(h C ) = h C , where h is the Lie algebra of 𝐻. Now Ad(𝑒 2𝑖𝑋 ) = exp(2𝑖ad(𝑋)). Since 𝑋 is in u(𝑛), 2𝑖ad(𝑋) is diagonalizable with real eigenvalues. Thanks to lemma 3.25, ad(𝑋)(h C ) ⊂ h C . Consequently, 𝑋 ∈ 𝑁 (h C ). Since 𝑋 is in u(𝑛), 𝑋 belongs to 𝑁 (h).
On the other hand, let 𝛾 in Γ. Then

𝑒 2𝑖𝑋 𝛾𝑒 -2𝑖𝑋 ∈ 𝐻 C since 𝑒 2𝑖𝑋 ∈ 𝑁 (𝐻 C ). Now we calculate 𝜄(𝑒 2𝑖𝑋 𝛾𝑒 -2𝑖𝑋 ) -1 (𝑒 2𝑖𝑋 𝛾𝑒 -2𝑖𝑋
) which on one side gives 𝑒 2𝑖𝑍 with 𝑍 ∈ h and on the other side, we have

𝜄(𝑒 2𝑖𝑋 𝛾𝑒 -2𝑖𝑋 ) -1 (𝑒 2𝑖𝑋 𝛾𝑒 -2𝑖𝑋 ) = (𝑒 -2𝑖𝑋 𝛾𝑒 2𝑖𝑋 ) -1 (𝑒 2𝑖𝑋 𝛾𝑒 -2𝑖𝑋 ) = 𝑒 -2𝑖𝑋 Ad(𝛾 -1 )(𝑒 4𝑖𝑋 )𝑒 -2𝑖𝑋 = 𝑒 -2𝑖𝑋 𝑒 4𝑖 Ad(𝛾 -1 )(𝑋) 𝑒 -2𝑖𝑋 .
Therefore, combining the two equalities together, we get

𝑒 2𝑖𝑍 = 𝑒 -2𝑖𝑋 𝑒 4𝑖 Ad(𝛾 -1 )(𝑋) 𝑒 -2𝑖𝑋 that is 𝑒 4𝑖 Ad(𝛾 -1 )(𝑋) = 𝑒 2𝑖𝑋 𝑒 2𝑖𝑍 𝑒 2𝑖𝑋
. Applying lemma 3.24 with 𝐴 = 2 Ad(𝛾 -1 )(𝑋) ∈ u(𝑛) and 𝐵 = 2𝑋 ∈ 𝑁 (h), we have finally Ad(𝛾 -1 )(𝑋) -𝑋 ∈ h, so 𝑋 -Ad(𝛾)(𝑋) ∈ h. This proves that 𝑋 belongs to Lie(𝑁 (𝐻)). Now, both 𝑒 𝑖𝑋 and 𝑘𝑒 𝑖𝑋 normalize 𝐻 C and so does 𝑘 ∈ 𝑁 (𝐻 C ). We have

𝑘𝐻 C 𝑘 -1 = 𝐻 C so 𝐻 = 𝐻 C ∩ U(𝑛) = 𝑘𝐻 C 𝑘 -1 ∩ U(𝑛) = 𝑘(𝐻 C ∩ U(𝑛))𝑘 -1 = 𝑘𝐻𝑘 -1
so 𝑘 belongs to 𝑁 𝐺 (𝐻). □

Proof of the main theorem

The main idea to prove theorem 1.1 is issued from the following erroneous proof. Let {𝐽 1 , . . . , 𝐽 𝑁 } be a generating set for the invariant algebra R[𝑉 ] 𝐺 . Then this set is a separating set for (𝑉, 𝐺) (see the discussion after remark 2.14). For 1 ≤ 𝑘 ≤ 𝑁 , let 𝑗 𝑘 denote the restriction of 𝐽 𝑘 to 𝑉 𝐻 . The inclusion 𝑉 𝐻 ˓→ 𝑉 induces the morphism

R[𝑉 ] 𝐺 → R[𝑉 𝐻 ] 𝑁 (𝐻) , {𝐽 1 , . . . , 𝐽 𝑁 } ↦ → {𝑗 1 , . . . , 𝑗 𝑁 } .
Then, the set {𝑗 1 , . . . , 𝑗 𝑁 } is separating for (𝑉 𝐻 , 𝑁 (𝐻)). If theorem 2.15 was true for a real representation, we could conclude that 𝐹 is a generating set for the invariant field R(𝑉 𝐻 ) 𝑁 (𝐻) and this would achieve the proof. Unfortunately this is false. However, this argument can be modified to become a real proof.

Proof of theorem 1.

(C(𝑉 C )) 𝐺 C / / (C(𝑊 )) 𝐾 C (C(𝑉 C )) 𝐺 / / (C(𝑊 )) 𝐾 Moreover, C(𝑉 C ) = R(𝑉 ) ⊗ R C and C(𝑊 ) = C((𝑉 𝐻 ) C ) = R(𝑉 𝐻 ) ⊗ R C. Hence the map (R(𝑉 )) 𝐺 (R(𝑉 𝐻 )) 𝑁 𝐺 (𝐻)
is also surjective because every real rational invariant is in the image of a complex rational invariant, and therefore it is also the image of its real part. We now prove the algebraicity of strata. Thanks to proposition 3. In this appendix, we recall a few results we need about Lie groups, and add some folklore results on real analytic functions and real analytic manifolds.

We now focus on Lie groups, and recall the following classic formula.

Lemma A.1 (Baker-Campbell-Hausdorff formula). [12, §3.4] Let 𝐺 be Lie group and g its Lie algebra. Consider 𝑋, 𝑌 ∈ g. Then the solution 𝑍 ∈ g of 𝑒 𝑋 𝑒 𝑌 = 𝑒 𝑍 is a formal series in iterated commutators of 𝑋 and 𝑌

𝑍 = 𝑋 + 𝑌 + 1 2 [𝑋, 𝑌 ] + 1 12 [𝑋, [𝑋, 𝑌 ]] - 1 12 [𝑌, [𝑋, 𝑌 ]] + • • • ,
where "• • • " indicates terms involving higher commutators of 𝑋 and 𝑌 . If 𝑋 and 𝑌 are sufficiently small elements of the Lie algebra g, the series is convergent.

This formula implies that the multiplication of 𝐺 in exponential coordinates near the neutral element is real analytic. From this, one deduces immediately the following result.

Corollary A.2. Let 𝐺 be a Lie group. Then 𝐺 has a natural real analytic structure such that:

• The multiplication map is globally real-analytic Lemma A.3. Let 𝐿 be a Lie group with a finite number of connected components, 𝐾 be a closed subgroup of 𝐿, and assume that there exists ℓ in 𝐿 such that ℓ𝐾ℓ -1 ⊂ 𝐾. Then, ℓ normalizes 𝐾.

Proof. The automorphism Ad(ℓ) stabilizes k, so it is an automorphism of k. Therefore Ad(ℓ -1 ) stabilizes k so ℓ -1 𝐾 𝑒 ℓ ⊂ 𝐾 𝑒 .

Let 𝐾/𝐾 𝑒 be the set of connected components of 𝐾. Then 𝐾/𝐾 𝑒 is a finite group and Ad(ℓ) induces an action on 𝐾/𝐾 𝑒 . Let 𝑘 be a representative of an element of 𝐾/𝐾 𝑒 which is in the kernel of Ad(ℓ). Then ℓ𝑘ℓ -1 belongs to 𝐾 𝑒 so 𝑘 = ℓ -1 (ℓ𝛾ℓ -1 )ℓ ∈ 𝐾 𝑒 , which means that the class of 𝑘 is 0 in 𝐾/𝐾 𝑒 . Hence Ad(ℓ) is injective, and since 𝐾/𝐾 𝑒 is finite, Ad(ℓ) is a bijection. This implies that for any 𝑘 in 𝐾 there exists an element of the form ℓ𝑘 ′ ℓ -1 in the connected component of 𝑘. This means that there exists 𝑘 ′′ in 𝐾 𝑒 such that ℓ𝑘 ′ ℓ -1 = 𝑘 ′′ 𝑘 so we get

ℓ -1 𝑘ℓ = ℓ -1 ( (𝑘 ′′ ) -1 ℓ𝑘 ′ ℓ -1 ) ℓ = ( ℓ -1 (𝑘 ′′ ) -1 ℓ ) 𝑘 ′ ∈ 𝐾. □
We now recall a folklore results on propagation of local real analytic identities.

Lemma A.4. Let 𝑛 be a positive integer, let 𝑉 be an open and connected subset of R 𝑛 , and 𝑓 : 𝑉 → R a real analytic function. If there exists 𝑣 ∈ 𝑉 such that 𝑓 = 0 on a neighborhood of 𝑣 then 𝑓 = 0 on 𝑉 .

Proof. Let 𝑆 be the set of points 𝑥 in 𝑈 such that all partial derivatives of 𝑓 vanish at 𝑥. This set is closed, but since 𝑓 is real analytic it is also open. Since 𝑣 ∈ 𝑆, 𝑆 is nonempty so 𝑆 = 𝑉 . □ Corollary A.5. Let 𝑛 be a positive integer, and assume to be given the following data:

• 𝑉 is an open and connected subset of R 𝑛 ,

• 𝑍 is a real analytic manifold,

• 𝑌 is a (closed) real-analytic submanifold of 𝑍,

• 𝑓 : 𝑉 → 𝑍 is an analytic function. Assume that there exists a point 𝑣 in 𝑉 and a neighborhood 𝑈 𝑣 of 𝑣 in 𝑉 such that 𝑓 (𝑈 𝑣 ) ⊂ 𝑌 . Then 𝑓 (𝑉 ) ⊂ 𝑌 .

Proof. Let 𝑊 be the subset of points 𝑥 in 𝑉 such that there exists a neighborhood 𝑈 𝑥 of 𝑥 in 𝑉 such that 𝑓 (𝑈 𝑥 ) ⊂ 𝑌 . By definition, 𝑊 is open. Let 𝑥 be a point in 𝑊 ∩ 𝑉 . As 𝑌 is closed, 𝑓 (𝑥) ∈ 𝑌 . Let 𝜙 : 𝑈 𝑦 → R 𝑚 be a real analytic chart on a neighborhood 𝑈 𝑦 of 𝑦 in 𝑍, such that 𝑈 𝑦 ∩ 𝑌 is given by the equations 𝜙 1 = • • • = 𝜙 𝑟 = 0, and let 𝑈 𝑥 be a connected neighborhood of 𝑥 in 𝑉 such that 𝑓 (𝑈 𝑥 ) ⊂ 𝑈 𝑦 . Then, for 1 ≤ 𝑖, 𝜙 𝑖 ∘ 𝑓 is a real analytic function on 𝑈 𝑥 that vanishes on an open subset of 𝑈 𝑥 (since 𝑥 ∈ 𝑊 ). By lemma A.4, for 1 ≤ 𝑖 ≤ 𝑟, 𝜙 𝑖 ∘ 𝑓 vanishes in 𝑈 𝑥 , which means that 𝑓 (𝑈 𝑥 ) ⊂ 𝑌 , so 𝑥 ∈ 𝑊 . It follows that 𝑊 ∩ 𝑉 = 𝑊 , which means that 𝑊 is closed in 𝑉 . Since 𝑉 is connected, 𝑊 = 𝑉 and in particular 𝑓 (𝑉 ) ⊂ 𝑌 . □

We end this appendix with some classical results on totally real submanifolds. Let 𝑋 be a complex manifold. A real submanifold 𝑆 of 𝑋 is totally real if for any 𝑠 in 𝑆, T 𝑠 𝑆 is a totally real subspace of T 𝑠 𝑋, which means that T 𝑠 𝑆 ∩ 𝑖T 𝑠 𝑆 = {0}.

Lemma A.6. Assume that 𝑆 is a totally real submanifold of a complex manifold that is moreover real analytic. Then, around each point 𝑠 of 𝑆, there exists an holomorphic chart 𝜙 :

𝑈 𝑠 → C 𝑛 , where 𝑈 𝑠 is a neighborhood of 𝑠, such that 𝜙(𝑈 𝑠 ∩ 𝑆) = 𝜙(𝑈 𝑠 ) ∩ R 𝑘 .
Proof. Let 𝑓 : R 𝑘 → 𝑆 be a real analytic parametrization of 𝑆 defined in a neighborhood of 0 such that 𝑓 (0) = 𝑠. We can complexify 𝑓 to a map 𝐹 → C 𝑘 . Now, for each point 𝑧 in C 𝑘 near 0, 𝑑𝐹 0 (𝑢 + 𝑖𝑣) = 𝑑𝑓 0 (𝑢) + 𝑖𝑑𝑓 0 (𝑣). Since 𝑇 𝑠 𝑆 is totally real, we see that 𝑑𝐹 0 is injective and has rank 𝑘. It follows that 𝐹 parameterizes a complex manifold of 𝑋 containing 𝑆. The result follows. □ Corollary A.7. Let 𝑋 be a complex manifold and 𝑆 be a totally real and real analytic submanifold of 𝑋 of maximal dimension. Then the analytic Zariski closure of 𝑆 is the union of the connected components of 𝑋 which intersect 𝑆.

Proof. It is enough to deal with the case where 𝑋 is connected. Let 𝑍 be a complex analytic set containing 𝑆. For any point 𝑠 in 𝑆, let 𝑍 ′ be an irreducible component of 𝑍 containing 𝑠 and let us write locally 𝑍 ′ = ⋂ 𝑟 𝑖=1 {𝑓 𝑖 = 0} around 𝑠. According to lemma A.6, we can write locally the inclusion of 𝑆 in 𝑍 around 𝑠 as the inclusion of R 𝑁 in C 𝑁 near 0. Then, 𝑓 𝑖|R 𝑁 = 0. Since for any multi-index 𝐼 = {𝑖 1 , . . . , 𝑖 𝑁 } we have

𝜕 𝐼 𝑓 𝜕𝑧 𝐼 (0) = 𝜕 𝐼 𝑓 𝜕𝑥 𝐼 (0) = 0,
we see that 𝑓 𝑖 = 0, so 𝑍 ′ contains a neighborhood of 𝑠 in 𝑋. In this appendix, we provide for the reader's convenience a proof of theorem 2.6 for the linear case, which combines the ideas of the proofs presented in [10] and [38].

We first need a topological lemma, which allows to propagate connexity. • The second property is granted if and only if there exists a covering (𝑈 𝑖 ) 𝑖∈𝐼 of 𝑈 such that ∆ 𝑈 𝑖 is an isomorphism. We claim that these two properties imply that ∆ 𝑋 is also an isomorphism. Indeed, the injectivity is straightforward and for the surjectivity, we argue as follows: if 𝑠 is a global section of 𝑗 * Z 𝑈 on 𝑋, then for all 𝑖, we can write 𝑠 |𝑈 𝑖 = ∆ 𝑈 𝑖 (𝑡 𝑖 ), and therefore ∆ 𝑈 𝑖 ∩𝑈 𝑗 (𝑡 𝑖 -𝑡 𝑗 ) = 0. Since ∆ is a monomorphism, the (𝑡 𝑖 ) ′ 𝑠 glue together to a global section 𝑡 of Z 𝑋 such that ∆ 𝑋 (𝑡) = 𝑠.

Lemma

To conclude, we write Proof. The Lie algebra g of 𝐺 acts on 𝑉 , besides this action is given by a morphism from g to o(𝑉 ). The tangent space of 𝐺 • 𝑣 at 𝑣 is given by

H 0 (𝑈, Z 𝑈 ) = H 0 (𝑋, 𝑗 * Z 𝑈 ) ≃ H 0 (𝑋, Z 𝑋 ) ≃ Z,
and
𝐸 𝑣 = {𝑋 • 𝑣, 𝑋 ∈ g} Since 𝑋 acts by an element of o(𝑉 ), (𝑋 • 𝑣|𝑣) = 0 so 𝑣 is orthogonal to 𝐸 𝑣 . Besides, if 𝑔 is in 𝐺 𝑣 , 𝑤 is in 𝐸 ⊥ 𝑣 and 𝑋 is in g, we have (𝑔 • 𝑤|𝑋 • 𝑣) = (𝑤|𝑔 -1 • (𝑋 • 𝑣)) = (𝑤|Ad(𝑔 -1 )(𝑋) • 𝑣) = 0 so 𝑔 • 𝑤 belongs to 𝐸 ⊥ 𝑣 .
Hence 𝐸 ⊥ 𝑣 is stable by 𝐺 𝑣 . Let 𝒪 be an orbit in 𝑉 , and assume that 𝑤 is the point of 𝒪 that minimizes the distance to 𝑣. Then 𝑤 is a critical point for the function 𝑔 ↦ → ‖𝑔 -1 • 𝑤 -𝑣‖ 2 = ‖𝑔 • 𝑣 -𝑤‖ 2 , which means that for all 𝑋 in g, (𝑋 • 𝑣|𝑣 -𝑤) = 0. Since 𝑣 belongs to 𝐸 ⊥ 𝑣 , so does 𝑤. □ Proof of theorem 2.6, linear case. We argue by induction on the dimension on 𝑉 . The case 𝑉 = {0} is obvious. We now separate two different situations.

𝑖 = Ω 𝑖-1 ∩ (𝑉 𝛾 𝑖 ) 𝑐 . Then, each Ω 𝑖 is dense in Ω 𝑖-1 .
Besides, for any 𝑣 in Ω 𝑖-1 ∩ 𝑉 𝛾 𝑖 , if 𝐵 is a small ball centered in 𝑣, then we separate two cases.

•

If 1 ≤ 𝑖 ≤ 𝑝, then 𝐵 ∩ (𝑉 𝛾 𝑖 ) 𝑐 is connected. • If 𝑝 + 1 ≤ 𝑖 ≤ 𝑝 + 𝑞, 𝐵 ∩ (𝑉 𝛾 𝑖 ) 𝑐 is not connected because 𝑉 𝛾 𝑖 is an hyperplane, but
𝛾 𝑖 has order 2 in Γ. Hence 𝛾 𝑖 swaps the two connected components of 𝐵 ∩ (𝑉 𝛾 𝑖 ) 𝑐 , so 𝜋(𝐵 ∩ (𝑉 𝛾 𝑖 ) 𝑐 ) is connected. Hence it follows by induction using lemma B.1 that all 𝜋(Ω 𝑘 ) are connected. In particular, since

Ω 𝑝+𝑞 = 𝑈 Ω = 𝑈 ∩ Ω, 𝜋(𝑈 Ω ) is connected. Case 2: the representation 𝜌 : g → o(𝑉 ) is not trivial. Let 𝑊 = {𝑣 ∈ 𝑉 such that ∀𝑋 ∈ g, 𝑋 • 𝑣 = 0}
be the kernel of the representation 𝜌. Then, by hypothesis, 𝑊 is a proper vector subspace of 𝑉 . First we claim that 𝑊 has codimension at least 2. Indeed, if 𝑊 has codimension 1, then the line 𝑊 ⊥ is stable by the action of 𝐺 𝑒 . Since orthogonal transformations of a line are ±1, the action of 𝐺 𝑒 is trivial on this line, which implies that 𝑊 = 𝑉 and gives a contradiction. It follows from this that 𝜋(Ω ∩ 𝑊 𝑐 ) is connected.

Next we claim that points of 𝑊 are not principal. Indeed, if 𝑤 is in 𝑊 , 𝐺 𝑤 contains the identity component 𝐺 𝑒 of 𝐺. If 𝑤 was principal, then for any 𝑤 ′ close to 𝑤, the same property would hold for 𝐺 𝑤 ′ . This would imply that 𝑤 ′ is in 𝑊 , so 𝑊 would have nonempty interior. Hence we can replace Ω by Ω ∩ 𝑊 ⊥ without changing the conclusions, that is assume that Ω ∩ 𝑊 = ∅.

If 𝑣 is in Ω, thanks to lemma B. We can now conclude by applying lemma B.1 to the pair (𝜋(𝑈 Ω ), 𝜋(Ω)). For 𝑣 in Ω, we can consider the neighborhood Denote by T 𝑛 (R 3 ) the vector space of 𝑛-th order tensors on the Euclidean space R 3 . We consider the representation of the rotation group SO(3) on T 𝑛 (R 3 ) given by (using Einstein notation) where 𝑗(𝑛 𝑛 𝑛) denotes the antisymmetric matrix with entries

𝐺 • 𝐵 of 𝑣 in 𝑉 . Then 𝑈 Ω ∩ 𝐺 • 𝐵 = 𝑈 𝑣 so 𝜋(𝑈 Ω ) ∩ 𝜋(𝐺 • 𝐵) = 𝒫/𝐺 𝑣 is connected. Hence 𝜋(𝑈 Ω ) is connected. □ Chapter 3 O (
(𝜌 𝑛 (𝑔)T) 𝑖 1 ...𝑖𝑛 = 𝑔 𝑖 1 𝑗 1 𝑔 𝑖 2 𝑗 2 . . . 𝑔 𝑖𝑛𝑗𝑛 T 𝑗 1 ...𝑗𝑛 , 𝑔 ∈ SO(3), T ∈ T 𝑛 (R 3 ). ( 3
𝑗(𝑛 𝑛 𝑛) = Ö 0 -𝑛 𝑧 𝑛 𝑦 𝑛 𝑧 0 -𝑛 𝑥 -𝑛 𝑦 𝑛 𝑥 0 è . Given a subgroup Γ of O(3), set Γ + = {𝑔 ∈ Γ | det 𝑔 = 1} and Γ -= {𝑔 ∈ Γ | det 𝑔 = -1} .
Then Γ = Γ + ∪ Γ -and we have the following classification (see [43, chapter XIII section 9], [19, page 352]).

Type

I: A closed subgroup Γ of O(3) is of type I if Γ -= ∅, in which case Γ is a subgroup of SO (3)
SO(3) ⊕ Z 𝑐 2 , O(2) ⊕ Z 𝑐 2 , SO(2) ⊕ Z 𝑐 2 , Z 𝑚 ⊕ Z 𝑐 2 , D 𝑚 ⊕ Z 𝑐 2 , T ⊕ Z 𝑐 2 , O ⊕ Z 𝑐 2 , I ⊕ Z 𝑐 2 Type III: A closed subgroup Γ of O(3) is of type III if Γ -̸ = ∅ but -I / ∈ Γ. Then, Γ + is a subgroup of index 2 in the subgroup Γ = Γ + ∪ (-Γ -) of SO(3).
In that case, there exists

𝛾 ∈ Γ ∖ Γ + such that -Γ -= 𝛾Γ + and Γ = Γ + ∪ -𝛾Γ + .
Five representatives of conjugacy classes of subgroups of type III can be deduced

Z - 2𝑛 = Z 𝑛 ∪ (-r(𝑒 𝑒 𝑒 3 , 𝜋 𝑛 ))Z 𝑛 ∀𝑛 ≥ 1. (3.3) D 𝑧 𝑛 = Z 𝑛 ∪ (-r(𝑒 𝑒 𝑒 1 , 𝜋))Z 𝑛 ∀𝑛 ≥ 2.
(3.4)

D 𝑑 2𝑛 = D 𝑛 ∪ (-r(𝑒 𝑒 𝑒 3 , 𝜋 𝑛 ))D 𝑛 ∀𝑛 ≥ 1. (3.5) O -= T ∪ (-r(𝑒 𝑒 𝑒 3 , 𝜋 2
))T.

(3.6) O(2) -= SO(2) ∪ (-r(𝑒 𝑒 𝑒 1 , 𝜋))SO(2). (3.7)
In figure 3

Z - 4 = Z 2 ∪ (-r(𝑒 𝑒 𝑒 3 , 𝜋/2)Z 2 ) = {I, r(𝑒 𝑒 𝑒 3 , 𝜋), -r(𝑒 𝑒 𝑒 3 , 𝜋/2), -r(𝑒 𝑒 𝑒 3 , 3𝜋/2)} .
We propose, in Table 3.1, generators for type I and type III closed O(3)-subgroups.

We give now some useful decomposition of subgroups T, O and I. To do so, let us first introduce the subgroup

Z 𝑢 𝑢 𝑢 𝑛 := ≠ r Å 𝑢 𝑢 𝑢, 2𝜋 𝑛 ã∑ , (3.8)
where the axis ⟨𝑢 𝑢 𝑢⟩ generated by 𝑢 𝑢 𝑢 is said to be the primary axis of Z 𝑢 𝑢 𝑢 𝑛 . We have

Z 𝑛 = Z 𝑒 𝑒 𝑒 3 𝑛
and, for any 𝑔 ∈ SO(3),

𝑔Z 𝑛 𝑔 -1 = Z 𝑔𝑒 𝑒 𝑒 3 𝑛 .
In particular, Z 𝑛 is given by

Z 𝑛 = ß r Å 𝑒 𝑒 𝑒 3 , 2𝑘𝜋 𝑛 ã , 𝑘 = 0, . . . , 𝑛 -1 ™ , (3.9)
Type I subgroup Order Generators for 𝑖 ≥ 2. We propose now details on compositions of the subgroups T, O and I, with explicit axes for all cyclic and dihedral subgroups they contain.

Z 𝑛 , 𝑛 ≥ 2 𝑛 r (𝑒 𝑒 𝑒 3 , 2𝜋 /𝑛) D 𝑛 , 𝑛 ≥ 2 2𝑛 r (𝑒 𝑒 𝑒 3 , 2𝜋 /𝑛) , r(𝑒 𝑒 𝑒 1 , 𝜋) T 12 r(𝑒 𝑒 𝑒 3 , 𝜋), r(𝑒 𝑒 𝑒 1 , 𝜋), r(𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 , 2𝜋 /3) O 24 r(𝑒 𝑒 𝑒 3 , 𝜋 /2), r(𝑒 𝑒 𝑒 1 , 𝜋), r(𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 , 2𝜋 /3) I 60 r(𝑒 𝑒 𝑒 3 , 𝜋), r(𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 , 2𝜋 /3), r(𝑒 𝑒 𝑒 1 + 𝜑𝑒 𝑒 𝑒 3 , 2𝜋 /5) 𝜑 := (1 + √ 5)/2 SO(2) ∞ r(𝑒 𝑒 𝑒 3 , 𝜃), 𝜃 ∈ [0, 2𝜋] O(2) ∞ SO(2), r(𝑒 𝑒 𝑒 1 , 𝜋) Type III subgroup Z - 2 2 -r(𝑒 𝑒 𝑒 3 , 𝜋) Z - 2𝑛 , 𝑛 ≥ 2 2𝑛 -r (𝑒 𝑒 𝑒 3 , 𝜋 /𝑛) D 𝑑 2𝑛 , 𝑛 ≥ 2 4𝑛 -r (𝑒 𝑒 𝑒 3 , 𝜋 /𝑛) , r(𝑒 𝑒 𝑒 1 , 𝜋) D 𝑧 𝑛 , 𝑛 ≥ 2 2𝑛 r (𝑒 𝑒 𝑒 3 , 2𝜋 /𝑛) , -r(𝑒 𝑒 𝑒 1 , 𝜋) O - 24 -r(𝑒 𝑒 𝑒 3 , 𝜋 /2), -r(𝑒 𝑒 𝑒 2 -𝑒 𝑒 𝑒 3 , 𝜋) O(2) - ∞ SO(2), -r(𝑒 𝑒 𝑒 1 , 𝜋)
D 𝑛 = ß r Å 𝑒 𝑒 𝑒 3 , 2𝑘𝜋 𝑛 ã , r(b 𝑖 , 𝜋); 𝑘 = 0, . . . , 𝑛 -1, 𝑖 = 1, . . . , 𝑛 ™ , ( 3
First, we have (see [53] for instance)

T = 3 ⋃ 𝑖=1 Z 𝑒 𝑒 𝑒 𝑖 2 ∪ 4 ⋃ 𝑗=1 Z 𝑠 𝑠 𝑠𝑡 𝑗 3 (3.11)
with

𝑠 𝑠 𝑠 𝑡 1 = 𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 , 𝑠 𝑠 𝑠 𝑡 2 = 𝑒 𝑒 𝑒 1 -𝑒 𝑒 𝑒 2 -𝑒 𝑒 𝑒 3 , 𝑠 𝑠 𝑠 𝑡 3 = -𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 -𝑒 𝑒 𝑒 3 , 𝑠 𝑠 𝑠 𝑡 4 = -𝑒 𝑒 𝑒 1 -𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 .
For the cubic group, we have

O = 3 ⋃ 𝑖=1 Z 𝑒 𝑒 𝑒 𝑖 4 ∪ 4 ⋃ 𝑗=1 Z 𝑠 𝑠 𝑠𝑡 𝑗 3 ∪ 6 ⋃ 𝑘=1 Z 𝑎 𝑎 𝑎𝑐 𝑘 2 (3.12)
with vectors a 𝑘 given by

a 𝑐 1 = e 1 + e 2 , a 𝑐 2 = e 1 -e 2 , a 𝑐 3 = e 1 + e 3 (3.13) a 𝑐 4 = e 1 -e 3 , a 𝑐 5 = e 2 + e 3 , a 𝑐 6 = e 2 -e 3 .
Finally, we have

I = 6 ⋃ 𝑖=1 Z 𝑢 𝑢 𝑢 𝑖 5 ∪ 10 ⋃ 𝑗=1 Z 𝑣 𝑣 𝑣 𝑗 3 ∪ 15 ⋃ 𝑘=1 Z 𝑤 𝑤 𝑤 𝑘 2 (3.14)
Taking 𝜑 := (1+ √ 5) 5

to be the golden ratio, vectors 𝑢 𝑢 𝑢 𝑖 are obtained as centers of icosahedron faces ( [86, Figure 11]):

𝑢 𝑢 𝑢 1 := (1 + 3𝜑)𝑒 𝑒 𝑒 1 + (2 + 𝜑)𝑒 𝑒 𝑒 3 , 𝑢 𝑢 𝑢 2 := (2 + 𝜑)𝑒 𝑒 𝑒 1 + (1 + 3𝜑)𝑒 𝑒 𝑒 2 , 𝑢 𝑢 𝑢 3 := (2 + 𝜑)𝑒 𝑒 𝑒 2 -(1 + 3𝜑)𝑒 𝑒 𝑒 3 𝑢 𝑢 𝑢 4 := -(2 + 𝜑)𝑒 𝑒 𝑒 2 -(1 + 3𝜑)𝑒 𝑒 𝑒 3 , 𝑢 𝑢 𝑢 5 = (1 + 3𝜑)𝑒 𝑒 𝑒 1 -(2 + 𝜑)𝑒 𝑒 𝑒 3 , 𝑢 𝑢 𝑢 6 := (2 + 𝜑)𝑒 𝑒 𝑒 1 -(1 + 3𝜑)𝑒 𝑒 𝑒 2 .
Then vectors 𝑣 𝑣 𝑣 𝑗 are obtained from vertices of icosahedron:

𝑣 𝑣 𝑣 1 := 𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 , 𝑣 𝑣 𝑣 2 := 𝜑𝑒 𝑒 𝑒 1 + 1 𝜑 𝑒 𝑒 𝑒 2 , 𝑣 𝑣 𝑣 3 := 𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 -𝑒 𝑒 𝑒
𝑤 𝑤 𝑤 1 := 𝑒 𝑒 𝑒 1 + (𝜑 + 1)𝑒 𝑒 𝑒 2 + Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 2 := Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 + (𝜑 + 1)𝑒 𝑒 𝑒 3 𝑤 𝑤 𝑤 3 := (𝜑 + 1)𝑒 𝑒 𝑒 1 + Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 4 := 𝜑𝑒 𝑒 𝑒 1 , 𝑤 𝑤 𝑤 5 := (𝜑 + 1)𝑒 𝑒 𝑒 1 + Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 2 -𝑒 𝑒 𝑒 3 𝑤 𝑤 𝑤 6 := (𝜑 + 1)𝑒 𝑒 𝑒 1 - Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 7 := Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 1 -𝑒 𝑒 𝑒 2 + (𝜑 + 1)𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 8 := (𝜑 + 1)𝑒 𝑒 𝑒 1 - Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 2 -𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 9 := 𝑒 𝑒 𝑒 1 + (𝜑 + 1)𝑒 𝑒 𝑒 2 - Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 3 𝑤 𝑤 𝑤 10 := 𝜑𝑒 𝑒 𝑒 2 , 𝑤 𝑤 𝑤 11 := -𝑒 𝑒 𝑒 1 + (𝜑 + 1)𝑒 𝑒 𝑒 2 + Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 12 = 𝜑𝑒 𝑒 𝑒 3 𝑤 𝑤 𝑤 13 := - Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 1 + 𝑒 𝑒 𝑒 2 + (𝜑 + 1)𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 14 := -𝑒 𝑒 𝑒 1 + (𝜑 + 1)𝑒 𝑒 𝑒 2 - Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 3 , 𝑤 𝑤 𝑤 15 := - Å 1 + 1 𝜑 ã 𝑒 𝑒 𝑒 1 -𝑒 𝑒 𝑒 2 + (𝜑 + 1)𝑒 𝑒 𝑒 3 .
Finally, the subgroups SO(2) and O(2) are given by

SO(2) = {r(𝑒 𝑒 𝑒 3 , 𝜃), 𝜃 ∈ [0, 2𝜋]} (3.15)
and

O(2) = {r(𝑒 𝑒 𝑒 3 , 𝜃), r(b, 𝜋), 𝜃 ∈ [0, 2𝜋], b ∈ (𝑥𝑦) -plane} . (3.16)

Irreducible representations

The study of a representation of a compact Lie group is often made easier by considering its decomposition into a direct sum of irreducible representations (the notion of reducibility is introduced in section 1.2). In fact, any representation of a compact Lie group can be decomposed, non uniquely, into a direct sum of irreducible representations (proof can be found in [43, Chapter XII, section 2]). The irreducible representations of the groups SO(3) and O(3) are known (up to isomorphism). Explicit models for these irreducible representations are provided by the so-called harmonic tensor spaces, described below.

Harmonic tensors

Let S 𝑛 (R 3 ) be the subspace of totally symmetric tensors of order 𝑛1 . Contracting two indices 𝑖, 𝑗 on a totally symmetric tensor S does not depend on the particular choice of the pair (𝑖, 𝑗).

Thus, we can refer to this contraction without any reference to a particular choice of indices. We will denote this contraction as tr(S), which is a totally symmetric tensor of order 𝑛 -2 and is called the trace of S.

Definition 3.2.1. Let S ∈ S 𝑛 (R 3 ), S is called harmonic if tr(S) = 0. The space of 𝑛-th order harmonic tensors is denoted by

H 𝑛 (R 3 ) (or simply H 𝑛 ). H 𝑛 is a linear subspace of S 𝑛 (R 3 ) of dimension 2𝑛 + 1. Remark 3.2.2. A 𝑛-th order tensor T ∈ T 𝑛 (R 3 ) is a 𝑛-linear mapping T : R 3 × R 3 × . . . × R 3 → R (𝑥 𝑥 𝑥 1 , . . . , 𝑥 𝑥 𝑥 𝑛 ) ↦ → T(𝑥 𝑥 𝑥 1 , . . . , 𝑥 𝑥 𝑥 𝑛 ).
Thus, the subspace S 𝑛 (R 3 ) of totally symmetric tensors can be identified with the vector space 𝒫 𝑛 (R 3 ) of homogeneous polynomials of degree 𝑛. Following this correspondence, a traceless totally symmetric tensor H corresponds to an harmonic polynomial ℎ i.e. with vanishing Laplacian: ∆ℎ = 0. We denote by ℋ 𝑛 (R 3 ) the space of harmonic polynomials of degree 𝑛. The subspace H 𝑛 (R 3 ) is invariant under the action of SO(3) (a rotated harmonic tensor stays harmonic) and this follows from the fact that the space of harmonic polynomials ℋ 𝑛 is SO(3)-invariant (see [14, proposition 5.10]).

Theorem 3.2.3. [43,14] The subspace H 𝑛 (R

(H 𝑛 (R 3 ), 𝜌 𝑛 ) or (H #𝑛 (R 3
), ρ𝑛 ) for some integer 𝑛.

Isotropy classes for irreducible representations

The problem of finding isotropy classes for irreducible representations started in mechanics with the study of symmetry changes of a material, in the sixties. Thereafter, Michel [77,[START_REF] Michel | Symmetry defects and broken symmetry. configurations hidden symmetry[END_REF] and Sattinger [8] developed a new approach to solve the problem, in the specific case of SO(3)representations, using a convenient group-theoretic framework. Since then, several contributions have been made in this subject and been extended to O(3)-representations, especially from the mathematical point of view by Ihrig and Gobulitsky [53], and Chossat et al [19] who made several corrections on the results of [53]. In figures 3.2, 3. which is not true.

Characterization of the closed isotropy strata for

H 4 (R 3 )
In this section, we give a polynomial characterization of the closed isotropy strata of SO(3)representation 𝜌 𝑛 on the space of harmonic tensors of order 4. For this, we introduce the following symmetric second-order tensors d 2 , . . . , d 10 , for 𝐻 ∈ H 4 (R 3 ):

d 2 = tr 13 (H 2 ) d 3 = tr 13 (H 3 ) d 4 = d 2 2 d 5 = d 2 Hd 2 d 6 = d 3 2 d 7 = d 2 2 Hd 2 d 8 = d 2 2 H 2 d 2 d 9 = d 2 2 Hd 2 2 d 10 = d 2 2 H 2 d 2 2
(3.17) We consider as well the nine fundamental invariants:

𝐽 𝑘 := tr(d 𝑘 ), 𝑘 = 2, . . . , 10.
Denote by q the euclidean metric and 𝜀 the Levi-Civita tensor. We define the symmetric secondorder tensors

d ′ 2 := d 2 - 1 3 tr(d 2 )q, c 3 := H : d 2 , c 4 := H : c 3 , v 5 := 𝜀 : (d 2 c 3 ), v 6 := 𝜀 : (d 2 c 4 ). if d 2 is isotropic then d ′ 2 = 0 and 𝐻 ∈ Σ [SO(3)] ⊂ Σ [D 4 ] ,
if d 2 is transversely isotropic then we consider two cases: if Conversely, v 5 = v 6 = 0 and tr(d sentations, as mentioned in theorem 3.2.5. For instance, in 1996, Forte and Vianello [37] solved definitively the problem for the SO(3)-representation on the space of elasticity tensors (a set of fourth-order tensors). In that case, eight isotropy classes were found. Based on this method, the problem was solved for other constitutive laws. For example, 16 isotropy classes were obtained for the O(3)-representation on the space of Piezoelectricity tensors (a set of third-order tensors) [START_REF] Nye | Physical Properties of Crystals[END_REF][START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF][START_REF] Weller | Etude des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF][START_REF] Newnham | Properties of Materials: Anisotropy, Symmetry, Structure[END_REF][START_REF] Zou | Symmetry types of the piezoelectric tensor and their identification[END_REF]. Similar results were obtained for other constitutive tensor spaces [38,66].

H × d 2 ̸ = 0 then H ∈ Σ [D 4 ] ⊂ Σ [D 4 ] since tr(H × d 2 ) = 0. And if H × d 2 = 0 then H ∈ Σ [O(2)] ⊂ Σ [D 4 ] [
if tr(H × d 2 ) ̸ = 0 then H ∈ Σ [D 3 ] ⊂ Σ [D 3 ] since (H : d 2 ) × d 2 = 0 [91, corollary 9.6]. if tr(H × d 2 ) = 0 then -if H × d 2 = 0 then H ∈ Σ [O(2)] ⊂ Σ [D 3 ] -if H × d 2 ̸ = 0 then H ∈ Σ [D 4 ] . However, if H ∈ Σ [D 4 ] ⊂ Σ [D 4 ] then, using [2, proposition 4.9], 6𝐽 6 = -3𝐽 3 2 + 9𝐽 2 𝐽 4 + 20𝐽
2 × c 3 ) = 0 imply that H ∈ Σ [D 2 ] if d 2 × d 2 2 ̸ = 0 or c 3 × c 2 3 ̸ = 0 or d 2 × c 3 ̸ = 0. Now, if d 2 × d 2 2 = 0, c 3 × c 2 3 =
However, the Forte-Vianello approach requires rather fine calculations and reasoning to establish the classification. This complexity makes difficult its application to more involved situations, such as constitutive tensors of order greater than 4 or coupled constitutive laws involving a family of tensors [54,33]. A systematic way to calculate the isotropy classes was proposed by Chossat and Guyard in [18] using a binary operation between conjugacy classes. This operation was named the clips operation in [START_REF] Olive | Symmetry classes for even-order tensors[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF][START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF][START_REF] Olive | Symmetry classes in piezoelectricity from second-order symmetries[END_REF], where it was generalized and used to determine isotropy classes for reducible representations.

Clips operation

Chossat and Guyard [18] found the isotropy classes of a direct sum of two irreducible SO(3)representations. To do so, they introduced a binary operation on the conjugacy classes of closed SO(3)-subgroups that allows one to compute the set of isotropy classes 𝒥 (𝑉 ) of a direct sum

𝑉 = 𝑉 1 ⊕ 𝑉 2 of
[𝐻 1 ] ⊚ [𝐻 2 ] := { [𝐻 1 ∩ 𝑔𝐻 2 𝑔 -1 ], 𝑔 ∈ 𝐺 } .
This definition extends to two families (finite or infinite) ℱ 1 and ℱ 2 of conjugacy classes

ℱ 1 ⊚ ℱ 2 = ⋃ [𝐻 𝑖 ]∈ℱ 𝑖 𝑖∈{1,2} [𝐻 1 ] ⊚ [𝐻 2 ].
The following lemma states a central result that is useful to find the isotropy classes of a reducible representation once we know the isotropy classes of irreducible ones (see [

𝒥 (𝑉 1 ⊕ 𝑉 2 ) = 𝒥 (𝑉 1 ) ⊚ 𝒥 (𝑉 2 ).
Using this result, one can find the isotropy classes of any representation 𝑉 provided we know

1. a stable decomposition 𝑉 = 𝑊 1 ⊕ • • • ⊕ 𝑊 𝑟 ,
2. the isotropy classes of each representations 𝑊 𝑘 , 𝑘 ∈ {1, . . . , 𝑟},

the clips table of [𝐻

1 ] ⊚ [𝐻 2 ] for all conjugacy classes [𝐻 1 ], [𝐻 2 ]
of closed subgroups of 𝐺.

Clips operations between closed O(3)-subgroups

As recalled in section 3.1 (see also [53] for details), any closed O(3)-subgroup is either of type

[𝐻 1 ] ⊚ [𝐻 2 ⊕ Z 𝑐 2 ] = [𝐻 1 ] ⊚ [𝐻 2 ] and [𝐻 1 ⊕ Z 𝑐 2 ] ⊚ [𝐻 2 ⊕ Z 𝑐 2 ] = ([𝐻 1 ] ⊚ [𝐻 2 ]) ⊕ Z 𝑐 2 .
2. for every subgroup 𝐻 of SO(3) we have

[𝐻] ⊚ [O(3)] = {[𝐻]} and [1] ⊚ [𝐻] = {[1]} .

Clips operations between closed O(3)-subgroups of type I

The clips tables for SO(3)-subgroups can be found in [18, (3.16) for details on these subgroups): the table given by Chossat [18] is the correct one in this case

[D 𝑛 ] ⊚ [O(2)] = {[1], [D 𝑛 ], [D 2 ](if 𝑛 is even), [Z 2 ]} since Olive in [86] has [D 2 ]
if 𝑛 is odd instead of 𝑛 even. Indeed, the following intersection

D 𝑛 ∩ 𝑔O(2)𝑔 -1 = ß r Å 𝑒 𝑒 𝑒 3 , 2𝑘𝜋 𝑛 ã , r(b 𝑖 , 𝜋) ™ ∩ {r(𝑔𝑒 𝑒 𝑒 3 , 𝜃), r(𝑔b, 𝜋)} gives D 2 only if 𝑛 is even (take for instance 𝑔 = r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
) : 𝑔𝑒 𝑒 𝑒 3 = -𝑒 𝑒 𝑒 1 = b 1 and 𝑔b 1 = -𝑒 𝑒 𝑒 3 ).
Clips between [T] and [T]: the table of Olive [86, table 1] is the correct one for this case

[T] ⊚ [T] = {[1], [Z 2 ], [D 2 ], [Z 3 ], [T]} since Chossat [18, table 1] omits the conjugacy class [D 2 ]. Indeed, [D 2 ] ∈ [T] ⊚ [T] since the intersection T ∩ 𝑔T𝑔 -1 , for T = 3 ⋃ 𝑖=1 Z 𝑒 𝑒 𝑒 𝑖 2 ∪ 4 ⋃ 𝑗=1 Z 𝑠 𝑠 𝑠𝑡 𝑗 3 (3.11), gives D 2 for 𝑔 = r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
)

for example.

Clips between [T] and [O]: we follow the table given by Olive [START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF] for this case )

[T] ⊚ [O] = {[1], [Z 2 ], [D 2 ], [Z 3 ], [T]} since
for example.

Clips between [O] and [I]: we consider the table 1 of Olive [START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF] [O]

⊚ [I] = {[1], [Z 2 ], [Z 3 ], [D 3 ], [T]} since [D 2 ] / ∈ [O] ⊚ [I]; the intersection O ∩ 𝑔I𝑔 -1 , for I = 6 ⋃ 𝑖=1 Z 𝑢 𝑢 𝑢 𝑖 5 ∪ 10 ⋃ 𝑗=1 Z 𝑣 𝑣 𝑣 𝑗 3 ∪ 15 ⋃ 𝑘=1 Z 𝑤 𝑤 𝑤 𝑘 2 (3.14),
gives D 2 only if there exists three orthogonal vectors 𝑤 𝑤 𝑤 𝑘 , 𝑘 = 1, . . . , 15, that turn to 𝑒 𝑒 𝑒 1 , 𝑒 𝑒 𝑒 2 , 𝑒 𝑒 𝑒 3

for some rotation 𝑔. However, the only three orthogonal vectors 𝑤 𝑤 𝑤 𝑘 which match this condition are the vectrs of the triplet (𝑤 𝑤 𝑤 4 , 𝑤 𝑤 𝑤 10 , 𝑤 𝑤 𝑤 12 ), and in that case the intersection gives T (since 𝑔 is necessarily the identity rotation in that case).

Clips between [I] and [I]: the table given by Olive [START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF] is the correct one in this case

[I] ⊚ [I] = {[1], [Z 2 ], [Z 3 ], [D 3 ], [Z 5 ], [D 5 ], [I]}
since Chossat has the conjugacy class [T] which cannot be realized by any rotation using the intersection I ∩ 𝑔I𝑔 -1 (the only rotation that could realize T is the identity rotation which gives I instead).

Clips operations between type II and type III subgroups: results

We give, in tables 3.2 and 3.3, clips operations between type II and type III subgroups, where we have used the notations 𝑑 := gcd(𝑚, 𝑛), 𝑑 𝑘 := gcd(𝑘, 𝑛), 𝑑 ′ 𝑘 := gcd(𝑘, 𝑚),

𝒵(𝑛) := ⎧ ⎨ ⎩ [Z 2 ] if 𝑛 even [Z - 2 ] else and L O := [1], [Z 2 ], [D 𝑑 2 ], [Z - 2 ], [D 𝑧 2 ], [Z 𝑑 3 ], [D 𝑑 3 ], [D 𝑧 𝑑 3 ].
Remark 3.4.3. In the following tables we have used the conventions

[Z 1 ] := [1], [D 1 ] = [Z 2 ], [D 𝑧 1 ] = [Z - 2 ], [D 𝑧 2 ] = [D 𝑑 2 ]. ⊚ [Z - 2𝑛 ] [D 𝑧 𝑛 ] [D 𝑑 2𝑛 ] [Z𝑚 ⊕ Z 𝑐 2 ] [1], [Z - 2𝑑 ] if 𝑚 𝑑 even [1], [Z 𝑑 ] else [1], [Z 𝑑 ] if 𝑚 odd [1], [Z 𝑑 ], [Z - 2 ] else [1], [Z2], [Z - 2 ], [Z - 2𝑑 ] if 𝑚 𝑑 even [1], [Z2], [Z - 2 ], [Z 𝑑 ]
if 𝑚 even and 𝑚 𝑑 odd

[1], [Z 𝑑 ] else [D𝑚 ⊕ Z 𝑐 2 ] [1], [Z - 2𝑑 ], 𝒵(𝑛) if 𝑚 𝑑 even [1], [Z 𝑑 ], 𝒵(𝑛) else [1], [Z 𝑑 ], [Z 𝑑 2 ] [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑑 ] if 𝑚 even [1], [Z 𝑑 ] [Z 𝑑 2 ], [Z - 2 ], [D 𝑧 𝑑 ] else [1], [Z2], [D 𝑑 2 ], [Z - 2 ] [Z - 2𝑑 ], [D 𝑧 2 ], [D 𝑑 2𝑑 ] if 𝑚 𝑑 even [1], [Z2], [D2], [Z - 2 ] [D 𝑧 2 ], [Z 𝑑 ], [D 𝑑 ], [D 𝑧 𝑑 ]
if 𝑚 even and 𝑚 𝑑 odd

[1], [Z2], [Z - 2 ], [Z 𝑑 ], [D 𝑑 ], [D 𝑧 𝑑 ] else [O ⊕ Z 𝑐 2 ] [1], [Z2] [Z 𝑑 3 ], [Z4] if 4|𝑛 [1], [Z2] [Z 𝑑 3 ], [Z - 4 ]
if 𝑛 even and 4 ∤ 𝑛

[1], [Z - 2 ], [Z 𝑑 3 ] else [1], [Z 𝑑 2 ], [Z 𝑑 3 ], [Z 𝑑 4 ] [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑑 3 ], [D 𝑧 𝑑 4 ] L O , [Z - 4 ] [Z4], [D4], [D 𝑧 4 ] if 4|𝑛 L O , [Z - 4 ], [D 𝑑 4 ]
if 𝑛 even and 4 ∤ 𝑛

L O else [T ⊕ Z 𝑐 2 ] [1], [Z 𝑑 3 ], 𝒵(𝑛) [1], [Z - 2 ], [Z 𝑑 2 ], [Z 𝑑 3 ], [D 𝑧 𝑑 2 ] [1], [Z2], [Z - 2 ], [D 𝑑 2 ], [D 𝑧 2 ], [Z 𝑑 3 ] [I ⊕ Z 𝑐 2 ] [1], 𝒵(𝑛), [Z 𝑑 3 ], [Z 𝑑 5 ] [1], [Z 𝑑 2 ], [Z 𝑑 3 ] [Z 𝑑 5 ], [Z - 2 ], [D 𝑧 𝑑 2 ] [1], [Z2], [Z - 2 ], [D 𝑑 2 ], [D 𝑧 2 ],[Z 𝑑 3 ], [Z 𝑑 5 ] [SO(2) ⊕ Z 𝑐 2 ] [1], [Z - 2𝑛 ] [1], [Z - 2 ], [Z𝑛] [1], [Z2], [Z - 2 ], [Z - 2𝑛 ] [O(2) ⊕ Z 𝑐 2 ] [1], 𝒵(𝑛), [Z - 2𝑛 ] [1], [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑛 ] [1], [Z2], [Z - 2 ], [D 𝑑 2 ], [D 𝑧 2 ], [D 𝑑 2𝑛 ]
Table 3.2: Clips between type II and III O(3)-subgroups (part 1)

and we would risk to not cover all the groups resulting from this intersection. For this reason, we use a characterization of these conjugacy classes making use of the clips between type I subgroups that have already been calculated before (see subsection 3.4.1). But, before this, a preparatory lemma is needed.

Lemma 3.5.1. Let (𝐾 + , K) be a pair of subgroups of a group 𝐺 of index 2 and 𝐻 be a subgroup of 𝐺. Then, either

𝐾 + ∩ 𝐻 = K ∩ 𝐻, or the pair (𝐾 + ∩ 𝐻, K ∩ 𝐻) is of index 2.
Proof. Consider the following exact sequence

1 → 𝐾 + 𝑖 → K 𝑝 → K/𝐾 + ≃ Z 2 → 1 Then 𝑝( K ∩ 𝐻) is a subgroup of Z 2 . Therefore, either 𝑝( K ∩ 𝐻) = Z 2 and in this case (𝐾 + ∩ 𝐻, K ∩ 𝐻) is a pair of index 2 (considering the exact sequence 1 → 𝐾 + ∩ 𝐻 → K ∩ 𝐻 → Z 2 ) or 𝑝( K ∩ 𝐻) = 1 and then K ∩ 𝐻 = 𝐾 + ∩ 𝐻.
Theorem 3.5.2. Let Γ be a type III subgroup of O(3), built from a pair of SO(3)-subgroups (Γ + , Γ) of index 2 and let 𝐻 be a subgroup of SO(3). Let 𝐿 be a subgroup of O(3) such that

[𝐿] ∈ [Γ] ⊚ [𝐻 ⊕ Z 𝑐 2 ] then, either 1. 𝐿 is of type I and [𝐿] ∈ ([Γ + ] ⊚ [𝐻]) ∩ ([ Γ] ⊚ [𝐻]), 2.
or 𝐿 is of type III built from a pair (𝐿

+ , L) of index 2 such that [𝐿 + ] ∈ [Γ + ]⊚[𝐻] and [ L] ∈ [ Γ] ⊚ [𝐻].
Proof. First, remark that -I does not belong to the intersection between a type II and a type III subgroups and hence if

[𝐿] ∈ [Γ] ⊚ [𝐻 ⊕ Z 𝑐 2 ] then 𝐿 is either of type I or type III. Let [𝐿] ∈ [Γ] ⊚ [𝐻 ⊕ Z 𝑐 2 ] then [𝐿]
is given by the union (3.18)

𝐿 = Γ + ∩ 𝑔𝐻𝑔 -1 ∪ (-(𝛾Γ + ∩ 𝑔𝐻𝑔 -1 )), 𝑔 ∈ SO(3), 𝛾 ∈ Γ ∖ Γ + . If 𝛾Γ + ∩ 𝑔𝐻𝑔 -1 = ∅ then 𝐿 is of type I and 𝐿 = Γ + ∩ 𝑔𝐻𝑔 -1 = Γ ∩ 𝑔𝐻𝑔 -1 . Hence, [𝐿] ∈ ([Γ + ] ⊚ [𝐻]) ∩ ([ Γ] ⊚ [𝐻]). If 𝛾Γ + ∩ 𝑔𝐻𝑔 -1 ̸ = ∅ then 𝐿 is of type III and Γ + ∩ 𝑔𝐻𝑔 -1 ̸ = Γ ∩ 𝑔𝐻𝑔 -1 . Since (Γ + , Γ) is a
pair of index 2 then, by lemma 3.5.1, the pair

(Γ + ∩ 𝑔𝐻𝑔 -1 , Γ ∩ 𝑔𝐻𝑔 -1 ) is of index 2. Hence, there exists 𝜎 ∈ Γ ∩ 𝑔𝐻𝑔 -1 ∖ (Γ + ∩ 𝑔𝐻𝑔 -1 ) such that 𝐿 = Γ + ∩ 𝑔𝐻𝑔 -1 ∪ (-𝜎(Γ + ∩ 𝑔𝐻𝑔 -1 ))
Hence the result.

In the next subsections, we compute the clips operation between type II and type III subgroups of O(3) using theorem 3.5.2 which involves clips between type I O(3)-subgroups that have already been calculated in previous works ( [18,[START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF], see also remark 3.

Clips with Z -

2𝑛

First, let us recall that Z - 2𝑛 is built from the couple (Z 𝑛 , Z 2𝑛 ), where Z 𝑛 is given by (3.9), as in the equation (3.3) of section 3.1

Z - 2𝑛 = Z 𝑛 ∪ (-𝛾Z 𝑛 ), 𝛾 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑛 )
where we denote Z 1 := {I} when 𝑛 = 1.

Lemma 3.5.3. Let 𝑛 ≥ 1 and 𝑚 ≥ 2 be two integers and 𝑑 = gcd(𝑚, 𝑛).

Then [Z - 2𝑛 ] ⊚ [Z 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎨ ⎩ { [1], [Z - 2𝑑 ] } if 𝑚 𝑑 even, {[1], [Z 𝑑 ]} else.
Proof. We deduce from (3.18) that

Z - 2𝑛 ∩ (𝑔Z 𝑚 𝑔 -1 ⊕ Z 𝑐 2 ) = (Z 𝑛 ∩ 𝑔Z 𝑚 𝑔 -1 ) ∪ (-(𝛾Z 𝑛 ∩ 𝑔Z 𝑚 𝑔 -1 )), 𝛾 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑛
) .

In the case when 𝑔𝑒 𝑒 𝑒 3 and 𝑒 𝑒 𝑒 3 are not colinear, such group reduces to 1, so we suppose now that 𝑔𝑒 𝑒 𝑒 3 = ±𝑒 𝑒 𝑒 3 . We thus have to consider

(Z 𝑛 ∩ Z 𝑚 ) ∪ (-(𝛾Z 𝑛 ∩ Z 𝑚 ))
where Z 𝑛 ∩ Z 𝑚 = Z 𝑑 , and 𝛾Z 𝑛 ∩ Z 𝑚 is obtained by solving the equations of unknown 𝑘 1 , 𝑘 2 ∈ Z

2𝑘 1 + 1 𝑛 = 2𝑘 2 𝑚 . ( 3
(2𝑘 1 + 1)𝑑𝑚 1 = 2𝑘 2 𝑑𝑛 1 we get that 𝑘 ′ is odd. On one hand, 2𝑘 2 𝑚 = 2𝑝𝑘 ′ 2𝑝𝑑
and on the other hand,

2𝑘 1 +1 𝑛 = 𝑘 ′ 𝑛 1 𝑑𝑛 1 with 𝑘 ′ odd. If we write 𝑘 ′ = 2𝑘 + 1 such that 𝑘 ∈ N, then 𝛾Z 𝑛 ∩ Z 𝑚 = ß r Å 𝑒 𝑒 𝑒 3 , (2𝑘 + 1)𝜋 𝑑 ã™ .
In the following, recall that

𝒵(𝑛) := Z - 2𝑛 ⊚ (Z 2 ⊕ Z 𝑐 2 ) = ⎧ ⎨ ⎩ [Z 2 ] if 𝑛 even, [Z - 2 ] else.
(3.20) Lemma 3.5.4. Let 𝑛 ≥ 1 and 𝑚 ≥ 2 be two integers and 𝑑 = gcd(𝑚, 𝑛).

Then [Z - 2𝑛 ] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎨ ⎩ { [1], [Z - 2𝑑 ], 𝒵(𝑛) } if 𝑚 𝑑 even, {[1], [Z 𝑑 ], 𝒵(𝑛)} else.
Proof. Recall from (3.10) that

D 𝑚 = Z 𝑚 ∪ 𝑚 𝑖=1 Z b 𝑖 2 , Z b 𝑖 2 := {𝑒, r(b 𝑖 , 𝜋)}
where b 𝑖 are called the secondary axes of the subgroup D 𝑚 , with

b 1 = 𝑒 𝑒 𝑒 1 , b 𝑘 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑚 ) b 𝑘-1 , 𝑘 = 2, . . . , 𝑚.
We have

Z - 2𝑛 ∩ (𝑔D 𝑚 𝑔 -1 ⊕ Z 𝑐 2 ) = ( Z 𝑛 ∩ 𝑔D 𝑚 𝑔 -1 ) ⋃ ( -(𝛾Z 𝑛 ∩ (𝑔D 𝑚 𝑔 -1 )) ) .
The non-trivial cases take place for 𝑔𝑒 𝑒 𝑒 3 = ±𝑒 𝑒 𝑒 3 and the union is then deduced from lemma 3.5.3, and for 𝑔b 𝑖 = ±𝑒 𝑒 𝑒 3 in which case we get 𝒵(𝑛), hence the conclusion.

Lemma 3.5.5. Let 𝑛 ≥ 1, 𝑑 3 = gcd(3, 𝑛) and 𝑑 5 = gcd(5, 𝑛). We have

[Z - 2𝑛 ] ⊚ [T ⊕ Z 𝑐 2 ] = {[1], [Z 𝑑 3 ], 𝒵(𝑛)} , [Z - 2𝑛 ] ⊚ [I ⊕ Z 𝑐 2 ] = {[1], 𝒵(𝑛), [Z 𝑑 3 ], [Z 𝑑 5 ]} . [Z - 2𝑛 ] ⊚ [O ⊕ Z 𝑐 2 ] = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ {[1], [Z 2 ], [Z 𝑑 3 ], [Z 4 ]} if 𝑛 is even and 4|𝑛, { [1], [Z 2 ], [Z 𝑑 3 ], [Z - 4 ] } if 𝑛 is even but 4 ∤ 𝑛, { [1], [Z - 2 ], [Z 𝑑 3 ] } if 𝑛 is odd.
Proof. By (3.18), we have to consider

(Z 𝑛 ∩ (𝑔𝐻𝑔 -1 )) ∪ (-(𝛾Z 𝑛 ∩ (𝑔𝐻𝑔 -1 ))) for 𝐻 = T, O, I.
[Z - 2𝑛 ] ⊚ [SO(2) ⊕ Z 𝑐 2 ] = { [1], [Z - 2𝑛 ] } , [Z - 2𝑛 ] ⊚ [O(2) ⊕ Z 𝑐 2 ] = { [1], 𝒵(𝑛), [Z - 2𝑛 ]
} with 𝒵(𝑛) given by (3.20).

Clips with D 𝑧 𝑛

As explained in section 3.1, the subgroup D 𝑧 𝑛 is obtained for -𝛾 = -r(e 1 , 𝜋), so that Lemma 3.5.7. Let 𝑚, 𝑛 ≥ 2 be two integers and 𝑑 = gcd(𝑚, 𝑛). Then

D 𝑧 𝑛 = Z 𝑛 ∪-𝛾Z 𝑛 = Z 𝑛 ∪{-
[D 𝑧 𝑛 ] ⊚ [Z 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎨ ⎩ {[1], [Z 𝑑 ]} if 𝑚 is odd, { [1], [Z - 2 ], [Z 𝑑 ] } if 𝑚 is even.
Proof. From (3.18) we have to consider

D 𝑧 𝑛 ∩ (𝑔Z 𝑚 𝑔 -1 ⊕ Z 𝑐 2 ) = (Z 𝑛 ∩ 𝑔Z 𝑚 𝑔 -1 ) ∪ (-(𝛾Z 𝑛 ∩ 𝑔Z 𝑚 𝑔 -1 )), 𝛾 = r(e 1 , 𝜋),
where 𝛾Z 𝑛 = {r(b 1 , 𝜋), . . . , r(b 𝑛 , 𝜋)}. Here, the only non-trivial cases are obtained when

𝑔Z 𝑚 𝑔 -1 = Z 𝑛 (see lemma 3.5.3) or Z b 𝑖 2
, which leads directly to the result.

Lemma 3.5.8. Let 𝑚, 𝑛 ≥ 2 be two integers and 𝑑 = gcd(𝑚, 𝑛) and 𝑑 2 = gcd(2, 𝑛). Then

[D 𝑧 𝑛 ] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎨ ⎩ ¶ [1], [Z 𝑑 2 ], [Z 𝑑 ], [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑑 ] © if 𝑚 is even, { [1], [Z 𝑑 2 ], [Z 𝑑 ], [Z - 2 ], [D 𝑧 𝑑 ] } if 𝑚 is odd.
Proof. We apply theorem 3.5.2 with Γ + = Z 𝑛 and Γ = D 𝑛 . We have from [86, table 1]

[Z 𝑛 ] ⊚ [D 𝑚 ] = {[1], [Z 𝑑 2 ], [Z 𝑑 ]} and [D 𝑛 ] ⊚ [D 𝑚 ] = {[1], [Z 2 ], [D 2 ](if 𝑚 and 𝑛 even), [Z 𝑑 ], [D 𝑑 ]} Hence, the classes of [D 𝑧 𝑛 ] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([Γ + ] ⊚ [𝐻]) ∩ ([ Γ] ⊚ [𝐻]) = {[1], [Z 𝑑 2 ], [Z 𝑑 ]}
and the classes corresponding to type III subgroups are in the set

{ [Z - 2 ], [D 𝑧 𝑑 2 ]( if 𝑚 is even), [D 𝑧 𝑑 ]
} .

We can check that all the eventualities can occur by using (3.18):

(Z 𝑛 ∩ 𝑔D 𝑚 𝑔 -1 ) ∪ (-(𝛾Z 𝑛 ∩ 𝑔D 𝑚 𝑔 -1 ))
where 𝛾Z 𝑛 = {r(b 𝑖 , 𝜋), 𝑖 = 1, . . . , 𝑛} and

D 𝑚 = Z 𝑚 ∪ 𝑚 𝑗=1 Z b 𝑗 2 , we get [Z 𝑑 2 ]
for a rotation 𝑔 such that, ∃𝑗 = 1, . . . , 𝑚, 𝑔b 𝑗 = 𝑒 𝑒 𝑒 3 (take for instance 𝑔 = r

( 𝑒 𝑒 𝑒 3 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
) that turns only 𝑒 𝑒 𝑒 1 to 𝑒 𝑒 𝑒 3 and nothing else),

[Z 𝑑 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑒 𝑒 𝑒 3 (for instance 𝑔 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 3 ) ), [Z - 2 ]
for a rotation 𝑔 such that, ∃𝑗 = 1, . . . , 𝑚 and ∃𝑖 = 1, . . . , 𝑛, 𝑔b 𝑗 = b 𝑖 (for instance

𝑔 = r ( 𝑒 𝑒 𝑒 1 , 𝜋 3 ) ), [D 𝑧 𝑑 ]
for the identity rotation for instance,

[D 𝑧 𝑑 2 ]
if 𝑚 is even and in this case we can take 𝑔 = r

( 𝑒 𝑒 𝑒 2 , 𝜋 2
) for instance so that we have

𝑔𝑒 𝑒 𝑒 3 = 𝑒 𝑒 𝑒 1 = b 1 and 𝑔b 2 = b 2 = 𝑒 𝑒 𝑒 2 .
Lemma 3.5.9. Let 𝑛 ≥ 2 be an integer, 𝑑 2 = gcd(2, 𝑛) and 𝑑 3 = gcd(3, 𝑛). We have

[D 𝑧 𝑛 ] ⊚ [T ⊕ Z 𝑐 2 ] = { [1], [Z - 2 ], [Z 𝑑 2 ], [Z 𝑑 3 ], [D 𝑧 𝑑 2 ]
} .

Proof. We deduce from [86, table 1] and theorem 3.5.2 that the classes in

[D 𝑧 𝑛 ] ⊚ [T ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([Z 𝑛 ] ⊚ [T]) ∩ ([D 𝑛 ] ⊚ [T]) = {[1], [Z 𝑑 2 ], [Z 𝑑 3 ]}
and the classes corresponding to type III subgroups are in the set

{ [Z - 2 ], [D 𝑧 𝑑 2 ]
} .

We can check that all the eventualities can occur by using (3.18):

(Z 𝑛 ∩ 𝑔T𝑔 -1 ) ∪ (-(𝛾Z 𝑛 ∩ 𝑔T𝑔 -1 ))
where

𝛾Z
[D 𝑧 𝑛 ] ⊚ [O ⊕ Z 𝑐 2 ] = { [1], [Z 𝑑 2 ], [Z 𝑑 3 ], [Z 𝑑 4 ], [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑑 3 ], [D 𝑧 𝑑 4 ]
} .

Proof. We deduce from [86, table 1] and theorem 3.5.2 that the classes in

[D 𝑧 𝑛 ] ⊚ [O ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([Z 𝑛 ] ⊚ [O]) ∩ ([D 𝑛 ] ⊚ [O]) = {[1], [Z 𝑑 2 ], [Z 𝑑 3 ], [Z 𝑑 4 ]}
and the classes corresponding to type III subgroups are in the set

{ [Z - 2 ], [D𝑑 2 𝑧 ], [D 𝑧 𝑑 3 ], [D 𝑧 𝑑 4 ]
} .

We can check that all the eventualities can occur by using (3.18):

(Z 𝑛 ∩ 𝑔O𝑔 -1 ) ∪ (-(𝛾Z 𝑛 ∩ 𝑔O𝑔 -1 ))
where We have

O = ⋃ 3 𝑖=1 Z 𝑒 𝑒
[D 𝑧 𝑛 ] ⊚ [I ⊕ Z 𝑐 2 ] = { [1], [Z 𝑑 2 ], [Z 𝑑 3 ], [Z 𝑑 5 ], [Z - 2 ], [D 𝑧 𝑑 2 ]
} .

Proof. We deduce from [86, table 1] and theorem 3.5.2 that the classes in

[D 𝑧 𝑛 ] ⊚ [I ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([Z 𝑛 ] ⊚ [I]) ∩ ([D 𝑛 ] ⊚ [I]) = {[1], [Z 𝑑 2 ], [Z 𝑑 3 ], [Z 𝑑 5 ]}
and the classes corresponding to type III subgroups are in the set

{ [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑑 3 ], [D 𝑧 𝑑 5 ]
} .

We can check that all the eventualities can occur, except [D 𝑧 𝑑 3 ] and [D 𝑧 𝑑 5 ], by using (3.18):

(Z 𝑛 ∩ 𝑔I𝑔 -1 ) ∪ (-(𝛾Z 𝑛 ∩ 𝑔I𝑔 -1 ))
where Lemma 3.5.12. For any integer 𝑛 ≥ 2, we have

I = ⋃ 6 𝑖=1 Z 𝑢 𝑢
[D 𝑧 𝑛 ] ⊚ [SO(2) ⊕ Z 𝑐 2 ] = { [1], [Z - 2 ], [Z 𝑛 ] } [D 𝑧 𝑛 ] ⊚ [O(2) ⊕ Z 𝑐 2 ] = { [1], [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑛 ]
} Proof. By theorem 3.5.2 we deduce that the classes in the clips

[D 𝑧 𝑛 ] ⊚ [SO(2) ⊕ Z 𝑐 2 ] are in the following list { [1], [Z - 2 ], [Z 𝑛 ] } .
All the classes of the above list can be realized by a rotation 𝑔 using the union Z 𝑛 ∩ 𝑔SO(2)𝑔 -1 ∪ (-(𝛾Z 𝑛 ∩ 𝑔SO(2)𝑔 -1 )) (3.18) where SO(2) consists of all the rotations around 𝑒 𝑒 𝑒 3 . Indeed, we get:

[Z - 2 ]
for a rotation 𝑔 such that, ∃𝑖 = 1, . . . , 𝑛, 𝑔𝑒 𝑒 𝑒 3 = b 𝑖 , [Z 𝑛 ] for the identity rotation for instance.

As for the classes in

[D 𝑧 𝑛 ] ⊚ [O(2) ⊕ Z 𝑐 2 ]
, by theorem 3.5.2 we know that such classes are in the following list

{ [1], [Z 𝑑 2 ], [Z - 2 ], [D 𝑧 𝑑 2 ], [D 𝑧 𝑛 ]
} .

By the same reasoning, the above classes can be all realized except for [Z 𝑑 2 ] using (3.18)

Z 𝑛 ∩ 𝑔O(2)𝑔 -1 ∪ (-(𝛾Z 𝑛 ∩ 𝑔O(2)𝑔 -1 ))
where 𝛾Z 𝑛 = {r(b 𝑖 , 𝜋), 𝑖 = 1, . . . , 𝑛} and 𝑔O(2)𝑔 -1 = {𝑟(𝑔𝑒 𝑒 𝑒 3 , 𝜃), 𝑟(𝑔b, 𝜋), b ∈ (𝑥𝑦) plan}. Indeed, we get

[Z - 2 ]
for a rotation 𝑔 such that, ∃𝑖 = 1, . . . , 3, 𝑔b = b 𝑖 (take for instance 𝑔 = r

( 𝑒 𝑒 𝑒 1 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 3 , 𝜋 2
) that turns only 𝑒 𝑒 𝑒 2 to 𝑒 𝑒 𝑒 1 ),

[D 𝑧 𝑑 2 ] for 𝑔 = r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
) for example, [D 𝑧 𝑛 ] for the identity rotation for example.

Clips with D 𝑑

2𝑛

First we have (see section 3.1)

D 𝑑 2𝑛 = D 𝑛 ∪ (-𝛾D 𝑛 ), 𝛾 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑛 )
where we can write Lemma 3.5.13. Let 𝑚, 𝑛 ≥ 2 be two integers and 𝑑 = gcd(𝑚, 𝑛). Then

D 𝑛 = ß r Å 𝑒 𝑒 𝑒 3 , 2𝑘 1 𝜋 𝑛 ã , r(b 2𝑙+1 , 𝜋); 𝑘 1 = 0, . . . , 𝑛 -1 ™ (3.21) -𝛾D 𝑛 = ß -r Å 𝑒 𝑒 𝑒 3 , (2𝑘 2 + 1)𝜋 𝑛 ã , -
[D 𝑑 2𝑛 ] ⊚ [Z 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ { [1], [Z 2 ], [Z - 2 ], [Z - 2𝑑 ] } if 𝑚 𝑑 even, { [1], [Z 2 ], [Z - 2 ], [Z 𝑑 ] } if 𝑚 even and 𝑚 𝑑 odd, {[1], [Z 𝑑 ]} else.
Proof. From (3.18) we have to consider intersection

D 𝑑 2𝑛 ∩ (𝑔Z 𝑚 𝑔 -1 ⊕ Z 𝑐 2 ) = (D 𝑛 ∩ 𝑔Z 𝑚 𝑔 -1 ) ∪ (-(𝛾D 𝑛 ∩ 𝑔Z 𝑚 𝑔 -1 )) 𝛾 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑛 )
which can always reduce to 1. Otherwise we only have to consider three cases: 𝑔e 3 = ±e 3 and we deduce intersection from lemma 3.5.3, 𝑔e 3 = ±b 2𝑙 for some 𝑙 and intersection reduces to Z - 2 for 𝑚 even, 𝑔e 3 = ±b 2𝑙+1 for some 𝑙 and intersection reduces to Z 2 for 𝑚 even, and this concludes the proof.

Lemma 3.5.14. Let 𝑚, 𝑛 ≥ 2 be two integers, 𝑑 = gcd(𝑚, 𝑛) and 𝑑 2 = gcd(2, 𝑛). Then

[D 𝑑 2𝑛 ] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ { [1], [Z 2 ], [D 𝑑 2 ], [Z - 2 ], [Z - 2𝑑 ], [D 𝑧 2 ], [D 𝑑 2𝑑 ] } if 𝑚 𝑑 is even, { [1], [Z 2 ], [D 2 ], [Z - 2 ], [D 𝑧 2 ], [Z 𝑑 ], [D 𝑑 ], [D 𝑧 𝑑 ] } if 𝑚 𝑑 is odd and 𝑚 is even, { [1], [Z 2 ], [Z - 2 ], [Z 𝑑 ], [D 𝑑 ], [D 𝑧 𝑑 ] } else.
Proof. We apply theorem 3.5.2 with Γ + = D 𝑛 and Γ = D 2𝑛 . We deduce from [86, table 1]

[D 𝑛 ] ⊚ [D 𝑚 ] = {[1], [Z 2 ], [D 2 ](if 𝑚 and 𝑛 even), [Z 𝑑 ], [D 𝑑 ]} and [D 2𝑛 ] ⊚ [D 𝑚 ] = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ {[1], [Z 2 ], [D 2 ], [Z 2𝑑 ], [D 2𝑑 ]} if 𝑚 𝑑 even {[1], [Z 2 ], [D 2 ], [Z 𝑑 ], [D 𝑑 ]} if 𝑚 even and 𝑚 𝑑 odd {[1], [Z 2 ], [Z 𝑑 ], [D 𝑑 ]} if 𝑚 odd Hence, the classes in [D 𝑧 𝑛 ] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([D 𝑛 ] ⊚ [D 𝑚 ]) ∩ ([D 2𝑛 ] ⊚ [D 𝑚 ]) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ {[1], [Z 2 ], [D 2 ]} if 𝑚 𝑑 even, {[1], [Z 2 ], [D 2 ], [Z 𝑑 ], [D 𝑑 ]} if 𝑚 even and 𝑚 𝑑 odd, {[1], [Z 2 ], [Z 𝑑 ], [D 𝑑 ]} if 𝑚 odd,
and the classes corresponding to type III subgroups are in the set

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ { [Z - 2 ], [D 𝑧 2 ], [Z - 2𝑑 ], [D 𝑧 2𝑑 ] } if 𝑚 𝑑 even, { [Z - 2 ], [D 𝑧 2 ], [D 𝑧 𝑑 ] } if 𝑚 even and 𝑚 𝑑 odd, { [Z - 2 ], [D 𝑧 𝑑 ] } if 𝑚 odd.
We can check that all the eventualities can occur by using (3.18):

(D 𝑛 ∩ 𝑔D 𝑚 𝑔 -1 ) ∪ (-(𝛾D 𝑛 ∩ 𝑔D 𝑚 𝑔 -1 ))
where D 𝑛 and 𝛾D 𝑛 are given in (3.22). We obtain

[D 𝑑 2𝑑 ] if
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [D 2 ] if 𝑚 and 𝑛 even, [D 𝑧 2 ] if 𝑚 even and 𝑛 odd, [Z 2 ]
if 𝑚 odd and 𝑛 even,

[Z - 2 ] if 𝑚 and 𝑛 odds,
for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = ±b 2𝑙+1 for some 𝑙,

[Z 2 ]
for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = ±b 2𝑙+1 for some 𝑙,

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ [D 𝑧 2 ] if 𝑚 even, [Z 2 ]
if 𝑚 odd and 𝑛 even,

[Z - 2 ] if
[D 𝑛 ] ⊚ [O] = {[1], [Z 2 ], [Z 𝑑 3 ], [Z 𝑑 4 ], [D 𝑑 2 ], [D 𝑑 3 ], [D 𝑑 4 ]}
where 𝑑 4 = 𝑔𝑐𝑑(4, 𝑛) and

[D 2𝑛 ] ⊚ [O] = {[1], [Z 2 ], [Z 𝑑 3 ], [Z 2𝑑 2 ], [D 2 ], [D 𝑑 3 ], [D 2𝑑 2 ]} .
We deduce, by theorem 3.5.2, that the classes in

[D 𝑑 2𝑛 ] ⊚ [O ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([D 𝑛 ] ⊚ [O]) ∩ ([D 2𝑛 ] ⊚ [O]) = {[1], [Z 2 ], [Z 𝑑 3 ], [D 𝑑 2 ], [D 𝑑 3 ], [Z 4 ](if 4 | 𝑛), [D 4 ](if 4 | 𝑛)}
and the classes corresponding to type III subgroups are in ¶

[Z - 2 ], [Z - 4 ]( if 𝑛 is even), [D 𝑧 2 ], [D 𝑧 𝑑 3 ], [D 𝑧 4 ](if 4 | 𝑛), [D 𝑑 4 ](if 𝑛 is even and 4 ∤ 𝑛) © .
We can check that all the eventualities can occur by using (3.18):

(D 𝑛 ∩ 𝑔O𝑔 -1 ) ∪ (-(𝛾D 𝑛 ∩ 𝑔O𝑔 -1 ))
where D 𝑛 and 𝛾D 𝑛 are given in (3.10) and (3.22) and O in (3.12). We get Lemma 3.5.17. For any integer 𝑛 ≥ 2 and 𝑑 𝑘 = gcd(𝑛, 𝑘) for 𝑘 = 2, 3, 5, we have

[D 𝑑 2𝑛 ] ⊚ [I ⊕ Z 𝑐 2 ] = { [1], [Z - 2 ], [Z 2 ], [D 𝑑 2 ], [D 𝑧 2 ], [Z 𝑑 3 ], [Z 𝑑 5 ]

}

Proof. We deduce from [86, table 1] and theorem 3.5.2 that the classes in

[D 𝑑 2𝑛 ] ⊚ [I ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([D 𝑛 ] ⊚ [I]) ∩ ([D 2𝑛 ] ⊚ [I]) = { [1], [Z 2 ], [D 𝑑 2 ], [Z 𝑑 3 ], [D 𝑧 𝑑 3 ], [Z 𝑑 5 ], [D 𝑧 𝑑 5 ]
} and the classes corresponding to type III subgroups are in the set

{ [Z - 2 ], [D 𝑧 2 ], [D 𝑧 𝑑 3 ], [D 𝑧 𝑑 5 ]
} .

We can check that all the eventualities can occur except

[D 𝑑 3 ], [D 𝑧 𝑑 3 ], [D 𝑑 5 ], [D 𝑧 𝑑 5 ]
for the same argument as in lemma 3.5.11 by using (3.18):

(D 𝑛 ∩ 𝑔I𝑔 -1 ) ∪ (-(𝛾D 𝑛 ∩ 𝑔I𝑔 -1 ))
where D 𝑛 and 𝛾D 𝑛 are given in (3.10) and (3.22) and I in (3.14). We get Finally, we have Lemma 3.5.18. For any integer 𝑛 ≥ 2 we have

[D 𝑑 2𝑛 ] ⊚ [SO(2) ⊕ Z 𝑐 2 ] = { [1], [Z 2 ], [Z - 2 ], [Z - 2𝑛 ] } [D 𝑑 2𝑛 ] ⊚ [O(2) ⊕ Z 𝑐 2 ] = ¶ [1], [Z 2 ], [Z - 2 ], [D 𝑑 2 ], [D 𝑧 2 ], [D 𝑑 2𝑛 ]
© .

Proof. By theorem 3.5.2 we deduce that the classes in the clips

[D 𝑑 2𝑛 ] ⊚ [SO(2) ⊕ Z 𝑐 2 ] are in the following list { [1], [Z 2 ], [Z - 2 ], [Z - 2𝑛 ]
} .

All the classes of the above list can be realized by a rotation 𝑔 using the union

D 𝑛 ∩ 𝑔SO(2)𝑔 -1 ∪ (-(𝛾D 𝑛 ∩ 𝑔SO(2)𝑔 -1 ))(3.18).
Indeed, we get:

[Z 2 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = b 2𝑙+1 , [Z - 2 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = b 2𝑙 , [Z - 2𝑛 ]
for the identity rotation for instance.

As for the classes in

[D 𝑑 2𝑛 ] ⊚ [O(2) ⊕ Z 𝑐 2 ]
, by theorem 3.5.2 we know that such classes are in the following list

¶ [1], [Z 2 ], [Z - 2 ], [D 𝑑 2 ], [D 𝑑 2𝑛 ], [D 𝑧 2 ]
© .

By the same reasoning, the above classes can be all realized by a rotation 𝑔 using (3.18)

D 𝑛 ∩ 𝑔O(2)𝑔 -1 ∪ (-(𝛾D 𝑛 ∩ 𝑔O(2)𝑔 -1 ))
where 𝑔O(2)𝑔 -1 = {𝑟(𝑔𝑒 𝑒 𝑒 3 , 𝜃), 𝑟(𝑔b, 𝜋), b ∈ 𝑥𝑦 plane}. We get,

[Z 2 ] for a rotation 𝑔 such that 𝑔 = r ( 𝑒 𝑒 𝑒 1 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
) that turns only 𝑒 𝑒 𝑒 3 to 𝑒 𝑒 𝑒 1 for instance,

O -= T ∪ (-𝛾T), 𝛾 = r ( 𝑒 𝑒 𝑒 1 , 𝜋 2
)

where

T = 3 ⋃ 𝑖=1 Z 𝑒 𝑒 𝑒 𝑖 2 4 ⋃ 𝑗=1 Z 𝑠 𝑠 𝑠𝑡 𝑗 3 , (3.24)
𝛾T = ß r (𝑎 𝑎 𝑎 𝑐 𝑘 , 𝜋) , r ( 𝑒 𝑒 𝑒 𝑖 , 𝜋 2
) , r Å 𝑒 𝑒 𝑒 𝑖 , 3𝜋 2 ã™ . ( 3
[O -] ⊚ [Z 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎨ ⎩ ¶ [1], [Z - 2 ], [Z 𝑑 ′ 3 ], [Z - 4 ] © if 4|𝑚, { [1], [Z 𝑑 ′ 2 ], [Z - 𝑑 ′ 2 ], [Z 𝑑 ′ 3 ] } otherwise.
Proof. From (3.18) we have to consider intersection

O -∩ (𝑔Z 𝑚 𝑔 -1 ⊕ Z 𝑐 2 ) = (T ∩ 𝑔Z 𝑚 𝑔 -1 ) ∪ (-(𝛾T ∩ 𝑔Z 𝑚 𝑔 -1 )) 𝛾 = r ( 𝑒 𝑒 𝑒 1 , 𝜋 𝑛 )
which can always reduce to 1. Otherwise we only have to consider three cases: ].

𝑔𝑒 𝑒 𝑒 3 =
Lemma 3.5.20. Let 𝑚 ≥ 2 be an integer. We have

[O -] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ¶ [1], [Z 2 ], [Z - 2 ], [Z 𝑑 ′ 3 ], [D 𝑧 𝑑 ′ 3 ], [Z - 4 ], [D 𝑧 2 ], [D 𝑑 4 ] © if 4|𝑚, ¶ [1], [Z 2 ], [Z - 2 ], [Z 𝑑 ′ 3 ], [D 𝑧 𝑑 ′ 3 ], [D 2 ], [D 𝑧 2 ] © if 𝑚 is even and 4 ∤ 𝑚, ¶ [1], [Z 2 ], [Z - 2 ], [Z 𝑑 ′ 3 ], [D 𝑧 𝑑 ′ 3 ] © if 𝑚 is odd.
Proof. We apply theorem 3.5.2 with Γ + = T and Γ = O. We deduce from [86, table 1]

[T] ⊚ [D 𝑚 ] = ¶ [1], [Z 𝑑 ′ 2 ], [Z 𝑑 ′ 3 ], [D 𝑑 ′ 2 ]
© and

[O] ⊚ [D 𝑚 ] = ¶ [1], [Z 2 ], [Z 𝑑 ′ 3 ], [Z 𝑑 ′ 4 ], [D 𝑑 ′ 2 ], [D 𝑑 ′ 3 ], [D 𝑑 ′ 4 ]
© .

Hence, the classes in

[O -] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([T] ⊚ [D 𝑚 ]) ∩ ([O] ⊚ [D 𝑚 ]) = ¶ [1], [Z 2 ], [Z 𝑑 ′ 3 ], [D 𝑑 ′ 2 ]
© and the classes corresponding to type III subgroups are in

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ¶ [Z - 2 ], [D 𝑑 4 ], [Z - 4 ], [D 𝑧 2 ], [D 𝑧 𝑑 ′ 3 ] © if 4 | 𝑚, ¶ [Z - 2 ], [D 𝑧 2 ], [D 𝑑 ′ 3 ] © if 𝑚 even and 4 ∤ 𝑚, ¶ [Z - 2 ], [D 𝑧 𝑑 ′ 3 ] © if 𝑚 odd.
We can check that all the eventualities can occur by using (3.18), except

[D 2 ] when 4 | 𝑚 since it becomes D 𝑑 4 in this case. Consider (T ∩ 𝑔D 𝑚 𝑔 -1 ) ∪ (-(𝛾T ∩ 𝑔D 𝑚 𝑔 -1 ))
where 𝛾T is given by (3.25). We obtain

[Z 2 ]
for a rotation 𝑔 such that 𝑔b 𝑖 = 𝑒 𝑒 𝑒 𝑗 for 𝑖 = 1, . . . , 𝑚 and 𝑗 = 1, . . . , 3 (take for instance

𝑔 = r ( 𝑒 𝑒 𝑒 2 , 𝜋 3
) ∘ r ( 𝑒 𝑒 𝑒 3 , 𝜋 2 ) ), [Z - 2 ]
for a rotation 𝑔 such that 𝑔b 𝑖 = 𝑎 𝑎 𝑎 𝑐 𝑘 for 𝑖 = 1, . . . , 𝑚 (take for instance 𝑔 = r Lemma 3.5.21. We have

( 𝑎 𝑎 𝑎 𝑐 𝑘 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 3 , 𝜋 4 ) ), [Z 𝑑 ′ 3 ]
[O -] ⊚ [T ⊕ Z 𝑐 2 ] = { [1], [Z 2 ], [Z - 2 ], [D 2 ], [D 𝑧 2 ], [Z 3 ], [T]
} .

Proof. We deduce from [86, table 1] and theorem 3.5.2 that the classes in

[O -] ⊚ [T ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([T] ⊚ [T]) ∩ ([O] ⊚ [T]) = {[1], [Z 2 ], [Z 3 ], [D 2 ], [T]}
and the classes corresponding to type III subgroups are in

{ [Z - 2 ], [D 𝑧 2 ]
} .

We can check that all the eventualities can occur by using (3.18):

(T ∩ 𝑔T𝑔 -1 ) ∪ (-(𝛾T ∩ 𝑔T𝑔 -1 )),
where 𝛾T is given by (3.25) and T in (3.11). We get

[Z 3 ] for a rotation 𝑔 such that 𝑔𝑠 𝑠 𝑠 𝑡 𝑗 = 𝑠 𝑠 𝑠 𝑡 𝑙 , [D 2 ] for a rotation 𝑔 = r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
) for instance,

[Z 2 ] for a rotation 𝑔 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
) for instance,

[D 𝑧 2 ]
for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 𝑗 for 𝑖 = 1, . . . , 3 and the two remaining 𝑒 𝑒 𝑒 𝑖 turn to two 𝑎 𝑎 𝑎 𝑐 𝑘 (for instance 𝑔 = r (

𝑒 𝑒 𝑒 3 , 𝜋 4 ) ), [Z - 2 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑎 𝑎 𝑎 𝑐 𝑘 for 𝑖 = 1, . . . 3 (for instance 𝑔 = r ( 𝑎 𝑎 𝑎 𝑐 1 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 3 , 𝜋 2 )
),

[T] for the identity rotation for instance.

Lemma 3.5.22. We have

[O -] ⊚ [O ⊕ Z 𝑐 2 ] = ¶ [1], [Z 2 ], [Z - 2 ], [Z 3 ], [Z - 4 ], [D 𝑧 2 ], [D 𝑧 3 ], [D 𝑑 4 ], [O -] © .
Proof. We deduce, by theorem 3.5.2, that the classes in

[O -] ⊚ [O ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([T] ⊚ [O]) ∩ ([O] ⊚ [O]) = {[1], [Z 2 ], [Z 3 ], [D 2 ], [T]}
and the classes corresponding to type III subgroups are in the set ¶

[Z - 2 ], [Z - 4 ], [D 𝑧 2 ], [D 𝑧 3 ], [D 𝑑 4 ], [O -] © .
We can check that all the eventualities can occur by using (3.18)

(T ∩ 𝑔O𝑔 -1 ) ∪ (-(𝛾T ∩ 𝑔O𝑔 -1 ))
where 𝛾T is given by (3.25) and O in (3.12):

[Z 2 ]
for a rotation 𝑔 such that 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = 𝑒 𝑒 𝑒 𝑖 for 𝑖 = 1, . . . , 3 (take for instance 𝑔 = r

( 𝑒 𝑒 𝑒 1 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 3 , -𝜋 4 ) ), [Z - 2 ] for a rotation 𝑔 = r ( 𝑎 𝑎 𝑎 𝑐 1 , 𝜋 3 ) ∘ r ( 𝑒 𝑒 𝑒 3 , 𝜋 2 ) , [D 𝑧 2 ]
[O -] ⊚ [I ⊕ Z 𝑐 2 ] = { [1], [Z 2 ], [Z - 2 ], [Z 3 ], [D 𝑧 2 ], [T]
} .

Proof. We deduce from [86, table 1] and theorem 3.5.2 that the classes in

[O -] ⊚ [I ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list where r(b, 𝜋) represent the symmetry with respect to all the axes in the 𝑥𝑦 plane.

([T] ⊚ [I]) ∩ ([O] ⊚ [I]) = {[1], [Z 2 ], [Z 3 ], [T]} [Z - 2 ]
Lemma 3.5.26. Let 𝑚 ≥ 2 be an integer and 𝑑 ′ 2 = gcd(2, 𝑚). We have

[O(2) -] ⊚ [Z 𝑚 ⊕ Z 𝑐 2 ] = { [1], [Z 𝑚 ], [Z - 𝑑 ′ 2 ]
} .

Proof. We get the result by considering the following intersection (from (3.18))

O(2) -∩ (𝑔Z 𝑚 𝑔 -1 ⊕ Z 𝑐 2 ) = (SO(2) ∩ 𝑔Z 𝑚 𝑔 -1 ) ∪ (-(𝛾SO(2) ∩ 𝑔Z 𝑚 𝑔 -1 ))
where 𝛾 = r(𝑒 𝑒 𝑒 1 , 𝜋) and 𝛾SO(2) = {r(b, 𝜋)}.

The proof of the following lemma is similar to the proof of lemma 3.5.8.

Lemma 3.5.27. For any integer 𝑚 ≥ 2, we have

[O(2) -] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ] = ⎧ ⎨ ⎩ { [1], [Z - 2 ], [D 𝑧 2 ], [D 𝑧 𝑚 ] } if 𝑚 is even, { [1], [Z 2 ], [Z - 2 ], [D 𝑧 𝑚 ] } if 𝑚 is odd.
Proof. We apply theorem 3.5.2 with Γ + = SO(2) and Γ = O(2). We deduce from [86, table 1]

[SO(2)] ⊚ [D 𝑚 ] = {[1], [Z 2 ], [Z 𝑚 ]} and [O(2)] ⊚ [D 𝑚 ] = ¶ [1], [Z 2 ], [D 𝑑 ′ 2 ], [D 𝑚 ] © . Hence, the classes in [O(2) -] ⊚ [D 𝑚 ⊕ Z 𝑐 2 ]
, corresponding to type I subgroups, are in the following list

([SO(2)] ⊚ [D 𝑚 ]) ∩ ([O(2)] ⊚ [D 𝑚 ]) = {[1], [Z 2 ]}
and the classes corresponding to type III subgroups are in the set ¶

[Z - 2 ], [D 𝑧 𝑑 ′ 2 ], [D 𝑧 𝑚 ] © .
We can check that all the eventualities can occur by using (3.18), except [D 2 ] when 4 | 𝑚 since it becomes D 𝑑 4 in this case. Consider

(SO(2) ∩ 𝑔D 𝑚 𝑔 -1 ) ∪ (-(𝛾SO(2) ∩ 𝑔D 𝑚 𝑔 -1 ))
where 𝛾SO(2) = {r(b, 𝜋), b ∈ 𝑥𝑦 plane}. We obtain

[Z 2 ]
for a rotation 𝑔 such that 𝑔b 𝑖 = 𝑒 𝑒 𝑒 3 (possible only if 𝑚 is odd since if 𝑚 is even the second part of the intersection will not be empty),

[Z - 2 ]
for a rotation 𝑔 such that 𝑔b 𝑖 = b for some b in the 𝑥𝑦 plane,

[D 𝑧 𝑑 ′ 2 ] for a rotation 𝑔 = r ( 𝑒 𝑒 𝑒 2 , 𝜋 2
) for example,

[D 𝑧 𝑚 ]
for the identity rotation for instance.

The argumentation for the calculation of the clips with O(2) -is very similar to the ones for D 𝑧 𝑛 exposed in subsection 3.5.2.

Lemma 3.5.28. We have

[O(2) -] ⊚ [T ⊕ Z 𝑐 2 ] = { [1], [Z 2 ], [Z 3 ], [Z - 2 ], [D 𝑧 2 ]
} .

[O(2

) -] ⊚ [O ⊕ Z 𝑐 2 ] = { [1], [Z 2 ], [Z - 2 ], [D 𝑧 2 ], [D 𝑧 3 ], [D 𝑧 4 ]
} .

[O(2

) -] ⊚ [I ⊕ Z 𝑐 2 ] = { [1], [Z 2 ], [Z - 2 ], [D 𝑧 2 ]
} .

And finally we deduce the clips with SO(2) ⊕ Z 𝑐 2 and O(2) ⊕ Z 𝑐 2 in the same way as in lemma 3.5.12. Lemma 3.5.29. We have

[O(2) -] ⊚ [SO(2) ⊕ Z 𝑐 2 ] = { [1], [Z - 2 ], [SO(2)] } . [O(2) -] ⊚ [O(2) ⊕ Z 𝑐 2 ] = { [D 𝑧 2 ], [O(2) -]
} .

Chapter 4

Application to constitutive laws

Introduction to Piezoelectricity coupled law

In mechanics, constitutive laws relate the stress imposed on a material and the resulting deformation at a macroscopic level. They are used to describe the behavior of a material exposed to some constraints. These laws are modeled using tensors and the questions related to their symmetries can be formulated in the language of group representation theory. We propose here to apply clips operation to the specific case of the linear Piezoelectricity law. We introduce the space of constitutive tensors occurring in the mechanical description of Piezoelectricity, which describes the electrical behavior of a material subject to mechanical stress. It is defined by a triplet of tensors given by an elasticity tensor, a piezoelectricity tensor and a permittivity tensor.

Such a space is naturally endowed with an O(3)-representation, and the finite set of isotropy classes is obtained in theorem 4.4.1 below.

We now recall the Piezoelectricity law (details can be found in [START_REF] Schouten | Tensor Analysis for Physicists[END_REF]64,[START_REF] Royer | Elastic waves in solids I: Free and guided propagation[END_REF]75]). First, the mechanical state of a material is characterized by two symmetric second order tensors: the stress tensor 𝜎 and the infinitesimal strain tensor 𝜀. The relation between these two tensors forms the constitutive law that describes the mechanical behavior of a specific material. In linear elasticity, the relation is linear known as the generalized Hooke's law, given by

𝜎 = E : 𝜀
where E is the elasticity tensor, a fourth order tensor having the following index symmetries

𝐸 𝑖𝑗𝑘𝑙 = 𝐸 𝑗𝑖𝑘𝑙 = 𝐸 𝑖𝑗𝑙𝑘 = 𝐸 𝑘𝑙𝑖𝑗 , 𝑖, 𝑗, 𝑘, 𝑙 ∈ {1, 2, 3} .
We define the associated space of elasticity tensors Ela, which is a 21-dimensional vector space.

Similarly to the mechanical state, the electrical state of a material is described by two vector fields: the electric displacement field 𝑑 𝑑 𝑑 and the electric field 𝑒 𝑒 𝑒. These two fields are related and the relation between them forms the constitutive law that describes the electrical behavior of a material. In the linear case, it is given by 𝑑 𝑑 𝑑 = S.𝑒 𝑒 𝑒 where the second order symmetric tensor S is called the permittivity tensor. We define S to be the vector space of the permittivity tensors, which is of dimension 6.

Finally, the Piezoelectricity law is given by the coupled law

⎧ ⎨ ⎩ 𝜎 = E : 𝜀 -𝑒 𝑒 𝑒.P
𝑑 𝑑 𝑑 = P : 𝜎 + S.𝑒 𝑒 𝑒 .

In components (using Einstein notation), we write

⎧ ⎨ ⎩ 𝜎 𝑖𝑗 = 𝐸 𝑖𝑗𝑘𝑙 𝜀 𝑘𝑙 -𝑒 𝑘 𝑃 𝑘𝑖𝑗 𝑑 𝑖 = 𝑃 𝑖𝑗𝑘 𝜎 𝑗𝑘 + 𝑆 𝑖𝑗 𝑒 𝑗 , 𝑖, 𝑗, 𝑘, 𝑙 ∈ {1, 2, 3}
which involves a third order tensor P called the piezoelectricity tensor, satisfying the index symmetry

𝑃 𝑖𝑗𝑘 = 𝑃 𝑖𝑘𝑗 , 𝑖, 𝑗, 𝑘 ∈ {1, 2, 3} .
The vector space of piezoelectricity tensors is an 18 dimensional vector space, noted Piez.

As a consequence, the linear electromechanical behavior of any homogeneous material is defined by a triplet 𝒫 of constitutive tensors 𝒫 := (E, P, S) ∈ Ela ⊕ Piez ⊕ S and we define 𝒫iez to be the space of Piezoelectricity constitutive tensors:

𝒫iez = Ela ⊕ Piez ⊕ S.
The

SO(3)-representation on the space of elasticity tensors

The elastic behavior of a material is represented by the elasticity tensor, the fourth order tensor E that relates the stress and strain tensors in the generalized Hooke law

𝜎 = E : 𝜀
where 𝜎 is the stress tensor and 𝜀 is the strain tensor and the operation ":" is the contraction of two indices (for instance, E : 𝜀 writes in components: (E : 𝜀) 𝑖𝑗 = 𝐸 𝑖𝑗𝑘𝑙 𝜖𝑘𝑙).

Elasticity tensor

The main mechanical object in this study is the elasticity tensor

E : S 2 (R 3 ) → S 2 (R 3 )
where S 2 (R 3 ) is the space of second order symmetric tensors on R 3 (equipped with the canonical basis 𝑒 𝑒 𝑒 1 , 𝑒 𝑒 𝑒 2 , 𝑒 𝑒 𝑒 3 ) for which we consider the following orthonormal basis (S

2 (R 3 ) = R 3 ⊗ R 3 ) 𝑒 11 = 𝑒 𝑒 𝑒 1 ⊗ 𝑒 𝑒 𝑒 1 𝑒 22 = 𝑒 𝑒 𝑒 2 ⊗ 𝑒 𝑒 𝑒 2 𝑒 33 = 𝑒 𝑒 𝑒 3 ⊗ 𝑒 𝑒 𝑒 3 𝑒 23 = 1 √ 2 (𝑒 𝑒 𝑒 2 ⊗ 𝑒 𝑒 𝑒 3 + 𝑒 𝑒 𝑒 3 ⊗ 𝑒 𝑒 𝑒 2 ) 𝑒 13 = 1 √ 2 (𝑒 𝑒 𝑒 1 ⊗ 𝑒 𝑒 𝑒 3 + 𝑒 𝑒 𝑒 3 ⊗ 𝑒 𝑒 𝑒 1 ) 𝑒 12 = 1 √ 2 (𝑒 𝑒 𝑒 1 ⊗ 𝑒 𝑒 𝑒 2 + 𝑒 𝑒 𝑒 2 ⊗ 𝑒 𝑒 𝑒 1 )
The elasticity tensor E is a contravariant fourth order tensor that takes a second order symmetric contravariant tensor 𝜎 and gives a second order symmetric covariant tensor 𝜀:

𝜎 𝑖𝑗 = 𝐸 𝑖𝑗𝑘𝑙 𝜀 𝑘𝑙 . (4.2)
Since we work in an orthonormal basis, we do not distinguish between contravariant and covariant tensors (we equip R 3 with the usual scalar product), hence equation (4.2) becomes

𝜎 𝑖𝑗 = 𝐸 𝑖𝑗𝑘𝑙 𝜀 𝑘𝑙 .
In the orthonormal basis of S(R 3 ) defined above, we write

{𝜎} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝜎 11 𝜎 22 𝜎 33 √ 2𝜎 23 √ 2𝜎 13 √ 2𝜎 12 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and {𝜀} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝜀 11 𝜀 22 𝜀 33 √ 2𝜀 23 √ 2𝜀 13 √ 2𝜀 12 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
The symmetry of the tensors 𝜎 and 𝜀 induces an indicial symmetry in the first two indices 𝑖, 𝑗 of 𝐸 𝑖𝑗𝑘𝑙 as well as in the last two indices 𝑘, 𝑙. Moreover, the fact that E derives from a potential induces a major symmetry in the indices so that in the end E is endowed with the following subscripts symmetries of its components (see [31,Chapter 3] or [67]),

𝐸 𝑖𝑗𝑘𝑙 = 𝐸 𝑗𝑖𝑘𝑙 = 𝐸 𝑖𝑗𝑙𝑘 = 𝐸 𝑘𝑙𝑖𝑗 .
We denote by Ela the space of elasticity tensors.

Thanks to these indicial symmetries, an elasticity tensor has 21 independent components and can be represented by a 6 × 6 symmetric matrix in the considered orthonormal basis of S(R 2 )

{E} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Isotropy classes for SO(3)-representation on Ela

Consider the action of SO(3) on the vector space Ela of elasticity tensors given by (using Einstein notation)

(𝑔 ⋆ E) 𝑖𝑗𝑘 = 𝑔 𝑖𝑝 𝑔 𝑗𝑞 𝑔 𝑘𝑟 𝑔 𝑙𝑠 𝐸 𝑝𝑞𝑟𝑠 , 𝑔 ∈ SO(3), E ∈ Ela.
The problem of classifying linear elastic materials according to their symmetries goes at least as far back as the work of Lord Kelvin ([107], [58]). From then, many researchers devoted a great effort to the problem especially in crystallographic theory since they took for granted that symmetries of elasticity tensors should be looked for within the set of crystal classes and they found seven groups of orthogonal tensors corresponding to seven crystal systems (for more details on crystallographic theory, one can refer to Backus [6], Rychlewski [START_REF] Rychlewski | On hooke's law[END_REF], Walpole [START_REF] Walpole | Fourth-rank tensors of the thirty-two crystal classes: multiplication tables[END_REF],

Cowin and Mehrabadi [24], Love [68], Federov [34] and many others). However, the requirements imposed by some symmetries that do not exist in crystal lattices could not be avoided which made the researchers in the domain settle for 10 symmetry classes for the elasticity tensors ( [45], [24], [START_REF] Schouten | Tensor analysis for physicists[END_REF], [START_REF] Yong-Zhong | On the completeness of the crystallographic symmetries in the description of the symmetries of the elastic tensor[END_REF]) until 1996 when Forte-Vianello ( [37]) proved that there exist exactly eight symmetry classes. Inspired by the observation of Khatkevich ([59]), Forte-Vianello clarified the mathematical problem about the symmetry classes of an elasticity tensor and removed the link with crystallographic point groups which was extremely confusing and lead to the false assumption that there were ten, rather than eight. Later, Chadwick and coauthors confirmed the eight classes in ( [17]).

Theorem 4.2.1. (Forte-Vianello 1996 [37]) There exists 8 isotropy classes for the SO(3)representation on the space of elasticity tensors Ela, given by

{[1], [Z 2 ], [D 2 ], [D 3 ], [D 4 ], [O], [O(2)], [SO(3)]} .

Harmonic decomposition of the space of elasticity tensors

The vector space Ela can be decomposed into a direct sum of irreducible spaces (see section 3.2).

Backus in [6] was the first to give such a decomposition to Ela:

Ela = H 0 (R 3 ) ⊕ H 0 (R 3 ) ⊕ H 2 (R 3 ) ⊕ H 2 (R 3 ) ⊕ H 4 (R 3 )
where H 𝑛 (R 3 ), for 𝑛 = 0, 2, 4, is the space of harmonic tensors of order 𝑛. Meaning that any E ∈ Ela can be written as

E = (𝛼, 𝛽, d ′ , v ′ , H), with 𝛼, 𝛽 ∈ H 0 (R 3 ) (called the isotropic components of E), d ′ , v ′ ∈ H 2 (R 3
) (the second-order harmonic components of E) and H ∈ H 4 (R 3 ) (the fourth-order harmonic component of E). The harmonic components 𝛼, 𝛽, d ′ and v ′ are given by

𝛼 = tr(d), 𝛽 = tr(v), d ′ := d - 1 3 tr(d) q, v ′ := v - 1 3 tr(v) q.
where d := tr 12 E (𝑑 𝑖𝑗 = 𝐸 𝑘𝑘𝑖𝑗 , using Einstein notation) and v := tr 13 E (𝑣 𝑖𝑗 = 𝐸 𝑘𝑖𝑘𝑗 ). In other words, an elasticity tensor E ∈ Ela admits the following explicit harmonic decomposition [6]:

E = E 𝑖𝑠𝑜 + q ⊗ (4) a + q ⊗ (2,2) b + H. (4.5)
where

E 𝑖𝑠𝑜 = 1 15 (𝛼 + 2𝛽) q ⊗ (4) q + 1 6 (𝛼 -𝛽) q ⊗ (2,2)
q, (4.6)

𝛼 = tr(d), 𝛽 = tr(v), (4.7)
and

a = 2 7 (d ′ + 2v ′ ), b = 2(d ′ -v ′ ). (4.8)
with d ′ and v ′ defined by (4.2.3).

In (4.5), q is the Euclidean canonical bilinear 2-form represented by the components (𝛿 𝑖𝑗 ) 𝑖,𝑗∈{1,2,3}

in any orthonormal basis and the tensor products ⊗

and ⊗

(2,2)

, between symmetric second-order tensors a, b, are defined as follows (using Einstein notation):

(a ⊗ (4) b) 𝑖𝑗𝑘𝑙 = 1 6 (𝑎 𝑖𝑗 𝑏 𝑘𝑙 + 𝑏 𝑖𝑗 𝑎 𝑘𝑙 + 𝑎 𝑖𝑘 𝑏 𝑗𝑙 + 𝑏 𝑖𝑘 𝑎 𝑗𝑙 + 𝑎 𝑖𝑙 𝑏 𝑗𝑘 + 𝑏 𝑖𝑙 𝑎 𝑗𝑘 ), and
(a ⊗ (2,2) b) 𝑖𝑗𝑘𝑙 = 1 6 (2𝑎 𝑖𝑗 𝑏 𝑘𝑙 + 2𝑏 𝑖𝑗 𝑎 𝑘𝑙 -𝑎 𝑖𝑘 𝑏 𝑗𝑙 -𝑎 𝑖𝑙 𝑏 𝑗𝑘 -𝑏 𝑖𝑘 𝑎 𝑗𝑙 -𝑏 𝑖𝑙 𝑎 𝑗𝑘 ).

4.3

O(3)-representation on the space of piezoelectricity tensors

To describe the electric state of a material, we use the piezoelectricity law. In fact, an electric displacement is induced in response to an applied mechanical stress and this property is modeled by a third order tensor P, called piezoelectricity tensor. The constitutive equation for this law is given by

𝑑 𝑑 𝑑 = P : 𝜎 (𝑑 𝑖 = 𝑃 𝑖𝑗𝑘 𝜎 𝑗𝑘 , 𝑖, 𝑗, 𝑘 ∈ {1, 2, 3})
where 𝑑 𝑑 𝑑 is the electric displacement vector and 𝜎 is the stress tensor.

The piezoelectricity tensor

The piezoelectricity tensor is a third order tensor

P : S 2 (R 3 ) → R 3
with indicial symmetry 𝑃 𝑖𝑗𝑘 = 𝑃 𝑖𝑘𝑗 , 𝑖, 𝑗, 𝑘 ∈ {1, 2, 3} ( [33]). We denote by Piez the space of piezoelectricity tensors.

A piezoelectricity tensor has 18 independent components and can be represented by a 3 × 6 matrix, either in so-called Voigt notation (to view the piezoelectricity parameters)

Isotropy classes for O(3)-representation on Piez

Consider the action of O(3) on the vector space Piez of piezoelectricity tensors given by

(𝑔 ⋆ P) 𝑖𝑗𝑘 = 𝑔 𝑖𝑝 𝑔 𝑗𝑞 𝑔 𝑘𝑟 𝑃 𝑝𝑞𝑟 , 𝑔 ∈ O(3), P ∈ Piez.
Obviously, the physical nature of the piezoelectric material is not affected by this transformation, only its constitutive tensors are transformed. The possible anisotropies of a constitutive law are modeled on the symmetry classes of the associated representation. Symmetry classes of the piezoelectricity tensors can be found in many works such as [START_REF] Nye | Physical Properties of Crystals[END_REF][START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF][START_REF] Weller | Etude des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF][START_REF] Newnham | Properties of Materials: Anisotropy, Symmetry, Structure[END_REF][START_REF] Zou | Symmetry types of the piezoelectric tensor and their identification[END_REF] who followed the approach of Forte-Vianello [37] in the case of the elasticity tensor. Later, Olive et al. proposed a general algorithm in [START_REF] Olive | Symmetry classes for even-order tensors[END_REF][START_REF] Olive | Symmetry classes for odd-order tensors[END_REF][START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF][START_REF] Olive | Symmetry classes in piezoelectricity from second-order symmetries[END_REF] to simplify the determination of symmetry classes based on the definition of clips operations on conjugacy classes (a strategy initiated by Chossat et al. [19,18] and defined in subsection 3.

: ¶ [1], [Z 2 ], [Z 3 ], [D 𝑧 2 ], [D 𝑧 3 ], [Z - 2 ], [Z - 4 ], [D 2 ], [D 3 ], [D 𝑑 4 ], [D 𝑑 6 ], [SO(2)], [O(2)], [O(2) -], [O -], [O(3)] © .

Harmonic decomposition of the space of piezoelectricity tensors

The harmonic decomposition of the space Piez for the O(3)-representation is given by

Piez ≃ H 3 ⊕ H #2 ⊕ H 1 ⊕ H 1 ,
where, if 𝑛 ∈ N, H 𝑛 denotes the space of 𝑛th-order harmonic tensors endowed with the standard representation and H #𝑛 refers to the same space endowed with the twisted representation (3.2) (see also [START_REF] Olive | Symmetry classes in piezoelectricity from second-order symmetries[END_REF]).

Any piezoelectricity tensor P ∈ Piez can be written as

P = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a, h) with 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 ∈ H 1 (R 3 ), a ∈ H #2 (R 3 ) and h ∈ H 3 (R 3 ).
Let ⊙ be the symmetric tensor product and P 𝑠 ∈ S 3 (R 3 ) denote the totally symmetric part of P (of components (P 𝑠 ) 𝑖𝑗𝑘 = 1 3 (𝑃 𝑖𝑗𝑘 + 𝑃 𝑗𝑖𝑘 + 𝑃 𝑘𝑗𝑖 )). Any piezoelectricity tensor P ∈ Piez can be decomposed as the sum

P = g + h where h := P 𝑠 - 3 5 q ⊙ tr(P 𝑠 ) ∈ H 3 (R 3 ), (4.11)
is the leading harmonic part of P, and

g := P -h = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a),
is orthogonal to h (i.e. ⟨g, h⟩ = 𝑔 𝑖𝑗𝑘 ℎ 𝑖𝑗𝑘 = 0).

Remark 4.3.2. The third-order tensors g = g(P) and h = h(P) are linear covariants of P.

Characterization of the cubic class

Here we give the characterization of the cubic class Σ [O -] for piezoelectricity tensors. This chapter is an ArXiv preprint [3] in which we solve the problem of the distance to isotropy strata using Lasserre's method for polynomial optimization problems. We take advantage of the characterization of the isotropy strata by polynomial equations in some cases (elasticity and piezoelectricity for instance) to formulate the problem of determining the closest tensor belonging to a specific isotropy stratum to an experimental one into a polynomial optimization problem. In other words, from a mechanical problem, we construct a minimization problem of a polynomial function under polynomial constraints. Lasserre provides a method for solving such problems that consists on building a hierarchy of relaxed semidefinite problems converging to the optimal minimum under some conditions. With this method, we were able to solve the problem for some numerical examples in cubic elasticity and cubic piezoelectricity for instance. Despite its efficiency in converging to the desired solution in a small amount of time with high numerical precision, this method is not always convenient. In fact, when the number of variables and the number of constraints increase, converging to the optimal solution becomes more difficult.

Introduction

In mechanics, linear constitutive laws are described by the orbit space of a representation of the three-dimensional orthogonal group on the vector space of the considered constitutive tensors T [45,13,21]. This orbit space is endowed with a natural stratification by isotropy classes [𝐻], the strata Σ [𝐻] being the set of tensors with symmetry group conjugate to 𝐻.

The symmetry group of a measured (raw) tensor T 0 is in general trivial. However, in practice, appealing to Curie principle-the symmetries of the causes are to be found in the effectsa symmetry of a constitutive tensor is often expected by observing the micro-structure of a material [5,24,25]. For instance, the elasticity tensor of a single crystal alloy with cubic crystal network is expected to be cubic ([O], see figure 1), the piezoelectric tensor of an aluminum nitride (AlN) alloyed with rocksalt transition metal nitrides is expected to become cubic ([O -]) for a high chromium concentration [49]. The mechanical problem thus comes down to the Date: July 9, 2022. 2020 Mathematics Subject Classification. 90C23; 14P10; 90C22; 74B05; 74E10. Key words and phrases. polynomial optimization; Lasserre's method; semidefinite programming; distance to a symmetry class; cubic symmetry; elasticity; piezoelectricity; semialgebraic and real algebraic geometry.

The authors were partially supported by CNRS Projet 80-Prime GAMM (Géométrie algébrique complexe/réelle et mécanique des matériaux). computation of the distance 𝑑(T 0 , Σ [𝐻] ) of a raw constitutive tensor T 0 to a closed isotropy stratum Σ [𝐻] .

In linear elasticity, which involves a fourth-order tensor E, the distance to an isotropy stratum has been formulated as the minimization problem [27,25,18,54,12] min

E∈Σ [𝐻] ‖E 0 -E‖, E = 𝜌 4 (𝑔)A, A ∈ Fix(𝐻), 𝑔 ∈ SO(3),
with the natural parameterization by normal form A (a fixed point set for a representative symmetry group 𝐻) and rotation 𝑔. This problem has, however, many local minima and several global minima, making the determination of all the solutions numerically difficult.

In this paper, we formulate the computation of the distance to an isotropy stratum as a polynomial optimization problem. To do so, we make use of the property that the closed isotropy strata Σ [𝐻] are basic closed semialgebraic sets [1,2,63,65]. For the fourth-order elasticity tensor, an explicit characterization of the closed strata Σ [𝐻] by polynomial equations and inequalities has recently been obtained, by means of polynomial covariants [61,Theorem 10.2]. Since such a result is not yet available in piezoelectricity, we have provided in theorem 7.2 a polynomial characterization of the cubic symmetry stratum Σ [O -] for the third-order piezoelectricity tensor.

We formulate the distance problems in question in such a way that we can apply a semialgebraic optimization method designed by Lasserre and coworkers [40,41,42,33,37] to compute directly the global miminum of a polynomial function over polynomial constraints describing a basic closed semialgebraic set. This method consists in building a sequence of semidefinite programs whose optimal values converge to the desired minimum, under some hypothesis on the constraints. The benefit is that there exist efficient algorithms to solve numerically semidefinite programs, based on methods used in linear programming [17,47,73], such as the ellipsoid method [30] or the interior point method [56,3,57,71,36,8,9,19]. The considered algorithm has been implemented by Lasserre and Henrion [33] in a Matlab freeware, named GloptiPoly [32], that aims to solve a sequence of relaxed semidefinite programs using SeDuMi (a Matlab toolbox for solving semidefinite programs created by Sturm [69,70]). Moreover, this algorithm, when its stopping criterion is satisfied, extracts (approximated) minimizers for the considered minimized function.

Organization of the paper. The paper is organized as follows. In section 2, we recall basic material on isotropy classes and we pose the problem of the distance to an isotropy class. In section 3, we introduce polynomial optimization and its formulation to semidefinite programs. In section 3 and section 4, we describe the Lasserre and coworkers method for solving polynomial optimization problems with semialgebraic constraints as well as the corresponding algorithm, implemented as the software GloptiPoly. As a direct application, we deal with three examples of constitutive tensors. In section 5, we illustrate the method with the academic example of the distance of a symmetric second-order tensor to the transversely isotropic stratum Σ [O(2)] . In section 6, we compute the distance of an experimental elasticity (fourth-order) tensor of a Nickel-based single crystal superalloy to the cubic stratum Σ [O] , and consequently we extract the cubic elasticity tensor the closest to the experimental one. Finally, in section 7, we detail how polynomial optimization allows to find the cubic piezoelectricity (third-order) tensors, in Σ [O -] , the closest to raw tensors for wurtzite alloys.

All the tensorial components will be expressed with respect to an orthonormal basis. Hence, no distinction will be made between covariant and contravariant components. The notation q = (𝛿 𝑖𝑗 ) stands for the Euclidean metric tensor.

Isotropy classes and strata -Distance to an isotropy stratum

Let 𝐺 be a compact group and 𝜌 : 𝐺 → GL(V) be a continuous representation of 𝐺 on a finite dimensional real vector space V. Given 𝑣 𝑣 𝑣 ∈ V, its orbit is the subset of V defined by Orb(𝑣 𝑣 𝑣) := {𝜌(𝑔)𝑣 𝑣 𝑣; 𝑔 ∈ 𝐺} , and its symmetry group (or isotropy group) is defined as 𝐺 𝑣 𝑣 𝑣 := {𝑔 ∈ 𝐺; 𝜌(𝑔)𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣} .

The concept of symmetry group allows to define an equivalence relation on V, which is coarser than the relation "to be in the same orbit" and defined as follows: two vectors 𝑣 𝑣 𝑣 1 and 𝑣 𝑣 𝑣 2 have the same isotropy class (or same symmetry class in mechanics [22,23]) if they have conjugate symmetry groups. In the following, we shall denote by [55,10,48]) that there is only a finite number of isotropy classes for any finite dimensional representation of a compact group. The set of conjugacy classes [𝐻] of closed subgroups of a compact group is endowed with a partial order relation (reflexivity and transitivity are direct and true even if 𝐺 is not compact but anti-symmetry requires the compacity of 𝐺 [11, Proposition 1.9]), given by

[𝐻] ⪯ [𝐾] ⇐⇒ ∃𝑔 ∈ 𝐺, 𝑔𝐻𝑔 -1 ⊂ 𝐾.
Due to the order relation defined on the conjugacy classes, we define a closed stratum to be the set consisting of vectors having at least the symmetry [𝐻], denoted by Σ [𝐻] , and defined by

Σ [𝐻] = {𝑣 𝑣 𝑣 ∈ V; [𝐻] ⪯ [𝐺 𝑣 𝑣 𝑣 ]} = ⋃ [𝐻]⪯[𝐾] Σ [𝐾] .
The isotropy stratum Σ [𝐻] and the closed isotropy stratum Σ [𝐻] are semialgebraic sets [1,2,63,65], i.e defined by polynomial equations and inequalities [14,9]. Actually, if 𝐺 is a subgroup of GL(V), we can give a direct proof of this fact. Indeed, if 𝐺 is a compact subgroup of GL(V), 𝐺 is a real algebraic set by [62,Chapter 3,paragraph 4,Theorem 5]. Notice that so is the subset 𝐺 𝑣 𝑣 𝑣 , if 𝑣 𝑣 𝑣 ∈ V, as it is described by polynomial equations in the coefficients of the matrices of the real algebraic set 𝐺. Now, since 𝐻 is a closed subgroup of 𝐺, 𝐻 is in particular a compact subgroup of GL(V) and then a real algebraic set as well. As a consequence, the closed isotropy stratum Σ [𝐻] = { 𝑣 𝑣 𝑣 ∈ V; ∃𝑔 ∈ 𝐺, ∀ℎ ∈ 𝐻, 𝑔ℎ𝑔 -1 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 } , and the isotropy stratum

Σ [𝐻] = { 𝑣 𝑣 𝑣 ∈ V; ∃𝑔 ∈ 𝐺, ∀ℎ ∈ 𝐻, 𝑔ℎ𝑔 -1 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 and ∃𝑔 ′ ∈ 𝐺, ∀𝑘 ∈ 𝐺, 𝑘𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 ⇒ 𝑔 ′ 𝑘𝑔 ′ -1 ∈ 𝐻 } ,
are both described by first-order formulae (in the sense of [9, Definition 2.2.3]) so that the sets Σ [𝐻] and Σ [𝐻] are semialgebraic sets by [9, Proposition 2.2.4] (the latter cited result is an avatar of Tarski-Seidenberg theorem which is an angular stone of semialgebraic geometry). We shall introduce the distance of a vector 𝑣 𝑣 𝑣 0 ∈ V to the closed isotropy stratum Σ [𝐻]

(1)

∆(𝑣 𝑣 𝑣 0 , Σ [𝐻] ) 2 := min 𝑣 𝑣 𝑣∈Σ [𝐻] ‖𝑣 𝑣 𝑣 0 -𝑣 𝑣 𝑣‖ 2 ,
for some 𝐺-invariant norm ‖•‖. A minimizer will be denoted by 𝑣 𝑣 𝑣 * .

Examples of interest for the present work are provided by Continuum Mechanics, for which 𝐺 is either SO (3) or O(3), V is a space of tensors on R 3 , endowed with the invariant norm

‖T‖ := √ 𝑇 𝑖 1 ...𝑖𝑛 𝑇 𝑖 1 ...𝑖𝑛 ,
and the action on a tensor T is written (in an orthonormal basis)

(𝜌(𝑔)T) 𝑖 1 ...𝑖𝑛 := 𝑔 𝑖 1 𝑗 1 . . . 𝑔 𝑖𝑛 𝑗𝑛 𝑇 𝑗 1 ...𝑗𝑛 , T ∈ V, 𝑔 ∈ 𝐺.
Finally, the O(3)-subgroups will be denoted according to the notations in [29].

Example 2.1. In elasticity,

V = Ela = S 2 (S 2 (R 3 )) = { E ∈ ⊗ 4 R 3 , 𝐸 𝑖𝑗𝑘𝑙 = 𝐸 𝑗𝑖𝑘𝑙 = 𝐸 𝑘𝑙𝑖𝑗 }
is the 21-dimensional vector space of elasticity tensors, and 𝐺 = SO(3). In that case there are exactly eight isotropy classes [1],

[Z 2 ], [D 2 ], [D 3 ], [D 4 ], [O(2)], [O], [SO(3)] [22]
, and the problem of the distance to an elasticity isotropy stratum has been investigated in [74,25,18,54,38,39].

Example 2.2. In piezoelectricity,

V = Piez = { e ∈ ⊗ 3 R 3 , e 𝑖𝑗𝑘 = e 𝑖𝑘𝑗 }
is the 18-dimensional vector space of piezoelectricity tensors, and 𝐺 = O(3). In that case there are exactly 16 isotropy classes [1], 58,76,60], and the problem of the distance to a piezoelectricity isotropy stratum has been investigated in [77].

[Z 2 ], [Z 3 ], [D 𝑧 2 ], [D 𝑧 3 ], [Z - 2 ], [Z - 4 ], [D 2 ], [D 3 ], [D 𝑑 4 ], [D 𝑑 6 ], [SO(2)], [O(2)], [O(2) -], [O -], [O(3)] [
When polynomial equations and/or inequalities characterizing the semialgebraic set Σ [𝐻] are known (see [6,61]), the distance to an isotropy stratum problem (1) reduces to minimize a polynomial function (the quadratic function ∆( • , Σ [𝐻] ) 2 ) under polynomial constraints. In that case, we can solve the distance to an isotropy stratum problem using polynomial and semialgebraic optimization [42,44,53,66], which allows to approximate numerically the global minimum of the function ∆( • , Σ [𝐻] ) 2 .

Semialgebraic optimization method

The problem of determining the constitutive tensor having a specific symmetry the closest to an experimental one can be viewed as an example of the problem of minimizing a polynomial function over polynomial constraints with 𝑔 1 , . . . , 𝑔 𝑚 ∈ R[𝑋] (see [14,9] for self-contained references on semialgebraic geometry). We now describe Lasserre's method [40,42], that will allow us to solve numerically the problem of the distance from an experimental tensor to a closed stratum. The method consists in constructing a sequence of semidefinite programs whose optimal values form a nondecreasing sequence which converges to the optimum 𝑓 * .

In this section, apart from theorem 4.1 which is a refinement of [42, Theorem 6.2], there is no original statement: we give the essential steps and results of the approach for pedagogical reasons and to be self-contained. For more details on Lasserre's method and the involved mathematical results and background, we refer to [42,44,66].

The first step of the method is to notice that the optimization problem (2) can be reformulated as

(3) 𝑓 * = inf {∫ R 𝑛 𝑓
𝑔 • 𝑦 := ⎛ ⎝ ∑ 𝛽∈N 𝑛 𝑔 𝛽 𝑦 𝛼+𝛽 ⎞ ⎠ 𝛼∈N 𝑛 ∈ R N 𝑛 .
Under an hypothesis called the Archimedean hypothesis, we can write the optimization problem (3) as ( 4)

𝑓 * = inf 𝑦∈R N 𝑛 { ⟨𝑓, 𝑦⟩; 𝑦 (0,...,0) = 1, 𝑀 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 = 0, . . . , 𝑚 }
where, if 𝑦 ∈ R N 𝑛 , ⟨𝑓, 𝑦⟩ := ∑ 𝛼 𝑓 𝛼 𝑦 𝛼 , and 𝑔 0 := 1. Here, if 𝑀 is a finite or infinite matrix with real coefficients, 𝑀 ⪰ 0 means that 𝑀 is positive semidefinite (an infinite symmetric matrix is called positive semidefinite if all its principal submatrices are positive semidefinite). The formulation ( 4) is a direct consequence of the following solution of the moment problem on 𝐾. This statement is a reformulation of [42, theorem 2.44] and is due to Putinar ([64]) and Jacobi-Prestel ( [35]). We recall the proof below but, first, we have to define the essential Archimedean hypothesis. We refer to [66, Theorem 1.1] (a result due to Schmüdgen) for a list of properties equivalent to the Archimedean hypothesis. Notice that if 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis, then 𝐾 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0} is necessarily compact. The crucial point is that, if 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis, then we have access to Putinar's Positivstellensatz. Theorem 3.3 (Putinar [64]). Suppose that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 describing 𝐾 satisfy the Archimedean hypothesis, and let

𝑝 ∈ R[𝑋]. If 𝑝(𝐾) ⊂]0; +∞[, then 𝑝 ∈ M(𝑔 1 , . . . , 𝑔 𝑚 ).
See also [66, section 2] and [44, section 3.7] for alternative proofs.

Proof of theorem 3.1. Let 𝑦 ∈ R N 𝑛 . The direct implication is actually true even if the polynomials 𝑔 1 , . . . , 𝑔 𝑚 do not satisfy the Archimedean hypothesis. Indeed, suppose that 𝑦 has a representing measure 𝜇 on 𝐾. Now, take any vector p = (𝑝 𝛼 ) 𝛼∈N 𝑛 of R N 𝑛 with finitely many nonzero coordinates and set 𝑝 := ∑ 𝛼 𝑝 𝛼 𝑋 𝛼 . If 𝑀 is any matrix, denote by 𝑀 𝑇 its transpose. If 𝑔 is any polynomial of {𝑔 0 , . . . , 𝑔 𝑚 }, we have then

p 𝑇 𝑀 (𝑔 • 𝑦) p = ∑ 𝛼,𝛽 𝑝 𝛼 𝑝 𝛽 ( ∑ 𝛾 𝑔 𝛾 ∫ R 𝑛 𝑥 𝑥 𝑥 𝛼+𝛽+𝛾 d𝜇(𝑥 𝑥 𝑥) ) = ∫ R 𝑛 𝑔(𝑥 𝑥 𝑥)𝑝(𝑥 𝑥 𝑥) 2 d𝜇(𝑥 𝑥 𝑥) ≥ 0.
By definition, for all 𝑥 𝑥 𝑥 ∈ 𝐾, 𝑔(𝑥 𝑥 𝑥) ≥ 0, and the support of 𝜇 is included in 𝐾.

Conversely, suppose that the matrices 𝑀 (𝑦), 𝑀 (𝑔 1 • 𝑦), . . . , 𝑀 (𝑔 𝑚 • 𝑦) of 𝑦 are positive semidefinite, and denote by 𝐿 𝑦 the linear mapping

𝑝 = ∑ 𝛼 𝑝 𝛼 𝑋 𝛼 ∈ R[𝑋] ↦ → ∑ 𝛼 𝑝 𝛼 𝑦 𝛼 ∈ R.
Let 𝑔 ∈ {𝑔 0 , . . . , 𝑔 𝑚 } and consider the symmetric bilinear form

R[𝑋] × R[𝑋] → R (𝑝, 𝑞) ↦ → 𝐿 𝑦 (
𝑦 𝛼 = 𝐿 𝑦 (𝑋 𝛼 ) = ∫ R 𝑛 𝑥 𝑥 𝑥 𝛼 d𝜇(𝑥 𝑥 𝑥)
i.e., 𝑦 is the moment sequence of the measure 𝜇.

□

From now on, we assume that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis so that we can write (5) 𝑓 * = inf 𝑦∈R N 𝑛 {⟨𝑓, 𝑦⟩; 𝑦 0 = 1, 𝑀 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 = 0, . . . , 𝑚} , where 𝑦 0 := 𝑦 (0,...,0) .

Lasserre's method to solve the optimization problem (4) consists, then, in relaxing this infinitedimensional problem into a sequence of finite-dimensional problems which are semidefinite programs. Semidefinite programs, or SDP's, are optimization problems over finite positive semidefinite symmetric matrices which generalize linear programs, and for which there exist efficient algorithms of numerical resolution. SDP-solving algorithms include methods inspired by the ones used in linear programming, such as interior point methods (see for instance the references [3,72,69,71,26,75]).

Below, we follow Lasserre's notations in [42, section 6.1

.1]. First, if 𝑘 ∈ N, let Λ(𝑘) := {(𝛼 1 , . . . , 𝛼 𝑛 ) ∈ N 𝑛 ; 𝛼 1 + • • • + 𝛼 𝑛 ≤ 𝑘} and, if 𝑦 ∈ R Λ(2𝑘) and 𝑘 ′ ∈ N satisfies 𝑘 ′ ≤ 𝑘, set 𝑀 𝑘 ′ (𝑦) := (𝑦 𝛼+𝛽 ) 𝛼,𝛽∈Λ(𝑘 ′ ) . If 𝑔 ∈ R[𝑋], set 𝑔 • 𝑦 := ⎛ ⎝ ∑ 𝛽∈N 𝑛 𝑔 𝛽 𝑦 𝛼+𝛽 ⎞ ⎠ 𝛼∈Λ(𝑘) ∈ R Λ(𝑘) .

⌉

, 𝑣 1 , . . . , 𝑣 𝑚 ) . For 𝑑 any integer such that 𝑑 ≥ 𝑑 0 , we then consider the optimization problem (6) 𝜌 𝑑 = inf

𝑦∈R Λ(2𝑑)
{⟨𝑓, 𝑦⟩; 𝑦 0 = 1, 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 = 0, . . . , 𝑚} , relaxed from (5).

For a given 𝑑 ≥ 𝑑 0 , the optimization problem ( 6) is a semidefinite program (and can then be numerically solved using SDP solvers). Indeed, for all 𝑦 ∈ R Λ(2𝑑) such that 𝑦 0 = 1 and all 𝑖 ∈ {0, . . . , 𝑚}, we can write

𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) = 𝐴 0 𝑖 + ∑ 𝛼∈Λ(2𝑑)∖{0} 𝑦 𝛼 𝐴 𝛼 𝑖
where, for all 𝛼 ∈ Λ(2𝑑), 𝐴 𝛼 𝑖 is a symmetric square matrix of size Λ(𝑑 -𝑣 𝑖 ) (see also [66, section 5]).

The following theorem of Lasserre ([42, Theorem 6.2], see also [66,Theorem 1.5]) asserts that the sequence of optima (𝜌 𝑑 ) 𝑑≥𝑑 0 converges to 𝑓 * : Theorem 3.4 (Lasserre). The sequence (𝜌 𝑑 ) 𝑑≥𝑑 0 is a nondecreasing sequence that converges to 𝑓 * .

Proof. Let 𝑑 be an integer such that 𝑑 ≥ 𝑑 0 and denote by 𝐹 𝑑 the set of vectors 𝑦 ∈ R Λ(2𝑑) such that 𝑦 0 = 1 and 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0 for all 𝑖 ∈ {0, . . . , 𝑚}. The set {⟨𝑓, 𝑦⟩; 𝑦 ∈ 𝐹 𝑑+1 } is included in the set {⟨𝑓, 𝑦⟩; 𝑦 ∈ 𝐹 𝑑 }. Indeed, if 𝑦 ∈ 𝐹 𝑑+1 and if we denote by 𝑦 the truncation (𝑦 𝛼 ) 𝛼∈Λ(2𝑑) of 𝑦, we have 𝑦 0 = 𝑦 0 = 1 and, for 𝑖 ∈ {0, . . . , 𝑚}, 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0 (because 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) is a principal submatrix of the positive semidefinite matrix 𝑀 𝑑+1-𝑣 𝑖 (𝑔 𝑖 • 𝑦)), as well as ⟨𝑓, 𝑦⟩ = ⟨𝑓, 𝑦⟩ since deg 𝑓 ≤ 2𝑑. As a consequence, 𝜌 𝑑 ≤ 𝜌 𝑑+1 .

We then show that the nondecreasing sequence (𝜌 𝑑 ) 𝑑≥𝑑 0 is bounded by 𝑓 * . Consider the formulation (5) of our optimization problem and denote by 𝐹 the set of sequences 𝑦 ∈ R N 𝑛 such that 𝑦 0 = 1 and 𝑀 (𝑔 𝑖 • 𝑦) ⪰ 0 for all 𝑖 ∈ {0, . . . , 𝑚}. Let 𝑦 be in 𝐹 and let 𝑦 := (𝑦 𝛼 ) 𝛼∈Λ(2𝑑) be the truncation of 𝑦. Again, we have 𝑦 0 = 1, 𝑀 𝑑-𝑣 𝑖 (𝑔 𝑖 • 𝑦) ⪰ 0, 𝑖 ∈ {0, . . . , 𝑚}, and ⟨𝑓, 𝑦⟩ = ⟨𝑓, 𝑦⟩, so that 𝜌 𝑑 ≤ 𝑓 * . Therefore, the sequence (𝜌 𝑑 ) 𝑑≥𝑑 0 converges.

The last step is to show that 𝑓 * is actually the limit of (𝜌 𝑑 ) 𝑑≥𝑑 0 . If 𝜖 is a positive real number, one can show that there exists 𝑑 ≥ 𝑑 0 such that 𝑓 * -𝜖 ≤ 𝜌 𝑑 ≤ 𝑓 * : the interested reader is invited to refer to [42,Theorem 6.2] or [66,Theorem 1.5]. The proof involves the dual SDP associated to (6), together with Putinar's Positivstellensatz 3.3. □

Lasserre's algorithm -GloptiPoly

The principle of Lasserre's algorithm to solve problem (2) is to numerically compute the sequence of optima (𝜌 𝑑 ) 𝑑≥𝑑 0 (which by theorem 3.4 converges to 𝑓 * ) using SDP solvers at each step. In order to complete this approach, one has to define a stopping criterion for the algorithm. In [42, section 6.1], Lasserre chooses a sufficient condition in terms of ranks of moment matrices, a condition which is motivated by the theorem below. The result we show is actually a slight generalization of [42,Theorem 6.6], that we decided to state in order to take into account the fact that a SDP solver, when applied to the SDP (6), only provides, at best, a numerical approximation of the optimum 𝜌 𝑑 . Let 𝜖 be a nonnegative real number, 𝑑 be an integer such that 𝑑 ≥ 𝑑 0 and denote 𝑣 := max(𝑣 1 , . . . , 𝑣 𝑚 ).

Theorem 4.1. Let 𝑦 ∈ 𝐹 𝑑 (we defined 𝐹 𝑑 in the proof of theorem 3.4) such that 𝜌 𝑑 ≤ ⟨𝑓, 𝑦⟩ ≤ 𝜌 𝑑 + 𝜖. If rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦) then 𝑓 * ≤ ⟨𝑓, 𝑦⟩ ≤ 𝜌 𝑑 + 𝜖 ≤ 𝑓 * + 𝜖.
Moreover, if we denote 𝑠 := rank 𝑀 𝑑 (𝑦), there exist at least 𝑠 points 𝑥 𝑥 𝑥 of 𝐾 such that 𝑓 * ≤ 𝑓 (𝑥 𝑥 𝑥) ≤ 𝑓 * + 𝜖.

In other words, if an optimal solution 𝑦, up to a fixed precision 𝜖, of the SDP (6) satisfies the above rank condition on its moment matrix, then ⟨𝑓, 𝑦⟩ is an approximation of 𝑓 * up to precision 𝜖. Furthermore, there exist at least rank 𝑀 𝑑 (𝑦) points of 𝐾 which are global minimizers of 𝑓 up to precision 𝜖. Remark 4.2.

(1) For 𝜖 = 0, we recover Theorem 6.6 in [42].

(2) The SDP solver used in the algorithm of Lasserre implemented in the freeware Glop-tiPoly 3 computes an element 𝑦 of 𝐹 𝑑 which is an approximation of an optimal solution of ( 6) and such that the rank 𝑟 of 𝑀 𝑑 (𝑦) is maximal among the ranks of moment matrices of elements of 𝐹 𝑑 . GloptiPoly then checks if the numerical rank of the principal submatrix 𝑀 𝑑-𝑣 (𝑦) of 𝑀 𝑑 (𝑦) is equal to 𝑟. The numerical rank of a matrix 𝑀 is, roughly speaking, the number of singular values of 𝑀 which are greater than a fixed precision, and the numerical rank of 𝑀 is not greater than its rank. As a consequence, if the numerical rank of 𝑀 𝑑-𝑣 (𝑦) is (at least) 𝑟, we have the inequalities 𝑟 ≤ rank 𝑀 𝑑-𝑣 (𝑦) ≤ rank 𝑀 𝑑 (𝑦) = 𝑟 so that rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦) and the stopping criterion of theorem 4.1 applies. More details about these questions can be found in [34, sections 4.4.1 and 4.4.2]. (3) In [42, section 6.1.2] is described the algorithm, implemented in GloptiPoly, which extract (approximated) global minimizers of 𝑓 when the rank condition is satisfied.

Theorem 4.1 is a consequence of the following one whose sketch of proof is postponed below. For any 𝑟 ∈ N ∖ {0}, a Borel measure 𝜇 on R 𝑛 is said to be 𝑟-atomic if there exist 𝑥 𝑥 𝑥 1 , . . . , 𝑥 𝑥 𝑥 𝑟 ∈ R 𝑛 and positive real numbers 𝜆 1 , . . . , 𝜆 𝑟 such that 𝜇 = ∑ 𝑟 𝑖=1 𝜆 𝑖 𝛿 𝑥 𝑥 𝑥 𝑖 . Theorem 4.3 (Curto-Fialkow [16], Laurent [43]). Let 𝑦 ∈ 𝐹 𝑑 . If rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦), then 𝑦 can be represented by a 𝑠-atomic measure, where 𝑠 := rank 𝑀 𝑑 (𝑦), whose support is included in 𝐾.

Proof of theorem 4.1. We adapt the proof of [42, theorem 6.6]. Suppose that rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦). Then, by theorem 4.3, 𝑦 has a 𝑠-atomic representing measure 𝜇 with support included in 𝐾 : there exist 𝑥 𝑥 𝑥 1 , . . . , 𝑥 𝑥 𝑥 𝑠 ∈ 𝐾 and 𝜆 1 , . . . ,

𝜆 𝑠 ∈]0; +∞[ such that 𝜇 = ∑ 𝑠 𝑖=1 𝜆 𝑖 𝛿 𝑥 𝑥 𝑥 𝑖 . In particular, since 𝑦 0 = 1, we have 1 = 𝑦 0 = ∫ R 𝑛 𝜇(𝑥 𝑥 𝑥) = ∑ 𝑠 𝑖=1 𝜆 𝑖 . Then ⟨𝑓, 𝑦⟩ = ∑ 𝛼∈Λ(2𝑑) 𝑓 𝛼 𝑦 𝛼 = ∫ R 𝑛 ∑ 𝛼∈Λ(2𝑑) 𝑓 𝛼 𝑥 𝑥 𝑥 𝛼 d𝜇(𝑥 𝑥 𝑥) = 𝑠 ∑ 𝑖=1 𝜆 𝑖 𝑓 (𝑥 𝑥 𝑥 𝑖 ) ≥ 𝑠 ∑ 𝑖=1 𝜆 𝑖 𝑓 * = 𝑓 * , so that 𝑓 * + 𝜖 ≥ 𝜌 𝑑 + 𝜖 ≥ ⟨𝑓, 𝑦⟩ ≥ 𝑓 * .
Finally suppose that there exists 𝑖 ∈ {1, . . . , 𝑠} such that 𝑓 (𝑥 𝑥 𝑥 𝑖 ) > 𝑓 * + 𝜖. This implies that

𝑠 ∑ 𝑗=1 𝜆 𝑗 𝑓 (𝑥 𝑥 𝑥 𝑗 ) > ⎛ ⎝ 𝑠 ∑ 𝑗=1 𝜆 𝑗 𝑓 * ⎞ ⎠ + 𝜖 = 𝑓 * + 𝜖,
which is not true according to the above inequalities. As a consequence, for all 𝑖 ∈ {1, . . . , 𝑠}, 𝑓 * ≤ 𝑓 (𝑥 𝑥 𝑥 𝑖 ) ≤ 𝑓 * + 𝜖. □

Proof of theorem 4.3. We point out the essential steps of the reasoning, referring to [44] for the detailed proofs. We have

rank 𝑀 𝑑-𝑣 (𝑦) ≤ rank 𝑀 𝑑-𝑣+1 (𝑦) ≤ • • • ≤ rank 𝑀 𝑑-1 (𝑦) ≤ rank 𝑀 𝑑 (𝑦)
and suppose that rank 𝑀 𝑑-𝑣 (𝑦) = rank 𝑀 𝑑 (𝑦): we obtain that rank 𝑀 𝑑-1 (𝑦) = rank 𝑀 𝑑 (𝑦). We can then recursively apply the Flat Extension Theorem 5.14 of [44] (originally due to Curto and Fialkow in [15]) to assert the existence of a sequence ̃ 𝑦 of R N 𝑛 such that, for all 𝛼 ∈ Λ(2𝑑),  𝑦 𝛼 = 𝑦 𝛼 and, for all 𝑘 ∈ N satisfying 𝑘 ≥ 𝑑 -𝑣, rank 𝑀 𝑘 (̃ 𝑦) = rank 𝑀 𝑑 (𝑦) = 𝑠.

In particular, for all 𝑘 ≥ 𝑑, since the principal submatrix 𝑀 𝑑 (𝑦) = 𝑀 𝑑 (̃ 𝑦) of 𝑀 𝑘 (̃ 𝑦) is positive semidefinite (because 𝑦 is in 𝐹 𝑑 ) and rank 𝑀 𝑘 (̃ 𝑦) = rank 𝑀 𝑑 (̃ 𝑦), the symmetric matrix 𝑀 𝑘 (̃ 𝑦) is also positive semidefinite (see [44,Definition 1.1]). In other words, the (infinite) moment matrix 𝑀 (̃ 𝑦) is positive semidefinite. Since, furthermore, rank 𝑀 (̃ 𝑦) = 𝑠, by [44, Theorem 5.1 (i)], there is a 𝑠-atomic measure 𝜇 representing ̃ 𝑦, and then 𝑦, with support the finite real algebraic set 𝑉 (𝐼) := {𝑥 𝑥 𝑥 ∈ R 𝑛 ; for all 𝑝 ∈ 𝐼, 𝑝(𝑥 𝑥 𝑥) = 0} where 𝐼 := {𝑝 ∈ R[𝑋]; 𝑀 𝑝 = 0} (the proof of Theorem 5.1 (i) of [44] involves real algebraic geometry).

The last step is then to prove that this support is included in 𝐾. Write 𝑉 (𝐼) = {𝑥 𝑥 𝑥 1 , . . . , 𝑥 𝑥 𝑥 𝑠 } and let 𝜆 1 , . . . , 𝜆 𝑠 ∈]0; +∞[ such that 𝜇 = ∑ 𝑠 𝑖=1 𝜆 𝑖 𝛿 𝑥 𝑥 𝑥 𝑖 . By Lemma 5.6 of [44], there exist 𝑝 1 , . . . , 𝑝 𝑠 ∈ R[𝑋] of degree at most 𝑑 -𝑣 such that, for all 𝑖, 𝑗 ∈ {1, . . . , 𝑠}, 𝑝 𝑖 (𝑥 𝑥 𝑥 𝑗 ) = 𝛿 𝑖𝑗 (see also [44,Lemma 2.3]). For all 𝑖 ∈ {1, . . . , 𝑠} and 𝑗 ∈ {1, . . . , 𝑚}, we then have, because

𝑦 ∈ 𝐹 𝑑 , 0 ≤ 𝑝 𝑇 𝑖 𝑀 𝑑-𝑣 (𝑔 𝑗 • 𝑦)𝑝 𝑖 = ∫ R 𝑛 𝑔 𝑗 (𝑥 𝑥 𝑥)𝑝 𝑖 (𝑥 𝑥 𝑥) 2 d𝜇(𝑥 𝑥 𝑥) = 𝑠 ∑ 𝑘=1 𝜆 𝑘 𝑔 𝑗 (𝑥 𝑥 𝑥 𝑘 )𝑝 𝑖 (𝑥 𝑥 𝑥 𝑘 ) 2 = 𝜆 𝑖 𝑔 𝑗 (𝑥 𝑥 𝑥 𝑖 )
(see the proof of proposition 3.1 above for the first equality) and, since 𝜆 𝑖 > 0, 𝑔 𝑗 (𝑥 𝑥 𝑥 𝑖 ) ≥ 0. As a consequence, for all 𝑖 ∈ {1, . . . , 𝑠}, 𝑥 𝑥 𝑥 𝑖 ∈ {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0} = 𝐾. □

We finally present the algorithm implemented by Lasserre and Henrion in the Matlab freeware GloptiPoly 3 to numerically solve polynomial optimization problems. For details on GloptiPoly and its use, see [33] and Appendix B of [42].

In order to solve a SDP relaxation 𝜌 𝑑 , 𝑑 ≥ 𝑑 0 , GloptiPoly 3 uses by default the SDP solver SeDuMi of Sturm [70]. Other SDP solvers can also be used as long as they are interfaced through Yalmip [46] (see section 5.9 of [33]).

The inputs of GloptiPoly are

• the variables 𝑋 1 , . . . , 𝑋 𝑛 , • the polynomial 𝑓 ∈ R[𝑋 1 , . . . , 𝑋 𝑛 ],
• the polynomials 𝑔 𝑗 ∈ R[𝑋 1 , . . . , 𝑋 𝑛 ], 𝑗 ∈ {1, . . . , 𝑚}, satisfying the Archimedean hypothesis, • a maximal relaxation order 𝑑 max ≥ 𝑑 0 .

The first output is a status number 𝜉 ∈ {-1, 0, 1}:

• 𝜉 = -1 means that the consider SDP solver could not solve numerically any of the relaxations 𝜌 𝑑 , 𝑑 ∈ {𝑑 0 , . . . , 𝑑 max }; • 𝜉 = 0 means that the solver numerically solved (that is up to a prescribed precision 𝜖) at least one of the relaxations 𝜌 𝑑 , 𝑑 ∈ {𝑑 0 , . . . , 𝑑 max }, but at each such success either no optimal solution was provided by the solver, either the rank stopping criterion of Theorem 4.1 was not satisfied by the (approximated up to precision 𝜖) obtained optimal solution 𝑦 𝑑 . In that case, the algorithm also outputs the last computed (and then greatest) optimal value 𝜌 𝑑 which is (up to precision 𝜖) a lower bound for 𝑓 * ; • 𝜉 = 1 means that the rank stopping criterion of Theorem 4.1 has been satisfied by an optimal solution 𝑦 𝑑 of a solved relaxation 𝜌 𝑑 , 𝑑 ∈ {𝑑 0 , . . . , 𝑑 max }. In that case, the algorithm also outputs 𝜌 𝑑 which is then an approximation of 𝑓 * up to the prescribed precision 𝜖. Lasserre's algorithm for polynomial optimization is, in pseudo code, the following (see [ (1) When the rank condition is satisfied, we can also ask GloptiPoly to extract minimizers up to precision 𝜖 (in the sense of Theorem 4.1), which involves the algorithm described in [42, section 6.1.2].

(2) If the output 𝜉 is 0 or -1, one can increase 𝑑 max to try to obtain an approximation (or a better lower bound) of 𝑓 * at a higher relaxation order. (3) There is no complexity known for Lasserre's method. Actually, we do not know if there is a maximal relaxation degree 𝑑 max , dependent on the inputs of the problem, which would ensure the rank stopping criterion to be satisfied at some ordre 𝑑 ≤ 𝑑 max . However, what makes this method advantageous is that it benefits from the interesting complexity of SDP solvers to solve semidefinite programs (see for instance [42] A.1.2).

We conclude this part by the following remark: the convergence of Lasserre's polynomial optimization method, described in the previous sections, takes place when the constraint set is a semialgebraic compact set satisfying the Archimedean property 3.2. Nevertheless, when the Archimedean condition is not satisfied but the polynomial function 𝑓 is coercive, Jeyakumar-Lasserre-Li in [37] provide a way to consider the optimization problem (2) as a problem with constraints satisfying the Archimedean condition : Lemma 4.5 (Jeyakumar-Lasserre-Li). Suppose that the polynomial function 𝑓 : R 𝑛 → R associated to 𝑓 ∈ R[𝑋 1 , . . . , 𝑋 𝑛 ] is coercive, and let 𝑐 > 0 and 𝑦 𝑦 𝑦 ∈ 𝐾 such that 𝑐 > 𝑓 (𝑦 𝑦 𝑦). Then the quadratic module M(𝑔 1 , . . . , 𝑔 𝑚 , 𝑐 -𝑓 ) associated to the semialgebraic set

 𝐾 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0, 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0} is Archimedean (in particular, ̃ 𝐾 is compact). Furthermore, 𝑓 * = inf 𝑥 𝑥 𝑥∈𝐾 𝑓 (𝑥 𝑥 𝑥) = inf 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥) = min 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥).
Proof. The set 𝐸 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0} is not empty since 𝑦 𝑦 𝑦 ∈ 𝐸. Furthermore, the set 𝐸 is compact. Indeed, if we suppose that 𝐸 is not bounded, we can find a sequence (𝑥 𝑛 ) 𝑛∈N of elements of 𝐸 such that ‖𝑥 𝑛 ‖ → +∞. But then 𝑓 (𝑥 𝑛 ) → +∞ since 𝑓 is coercive, which is impossible since, by definition of 𝐸, for all 𝑛 ∈ N, 𝑓 (𝑥 𝑛 ) ≤ 𝑐. Since 𝐸 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0} is compact, then the quadratic module M(𝑔 1 , . . . , 𝑔 𝑚 , 𝑐 -𝑓 ) is Archimedean by [66,Theorem 1.1]. Finally, we have 𝑐 > 𝑓 (𝑦 𝑦 𝑦) so 𝑐 > 𝑓 * and

𝑓 * = inf {𝑓 (𝑥 𝑥 𝑥); 𝑥 𝑥 𝑥 ∈ 𝐾} = inf {𝑓 (𝑥 𝑥 𝑥); 𝑥 𝑥 𝑥 ∈ 𝐾 ∩ {𝑐 -𝑓 ≥ 0} } = inf { 𝑓 (𝑥 𝑥 𝑥); 𝑥 𝑥 𝑥 ∈ ̃ 𝐾 } .

□

In other words, even if the polynomials 𝑔 1 , . . . , 𝑔 𝑚 do not satisfy themselves the Archimedean hypothesis, provided that 𝑓 is coercive, we can place ourselves in the range of application of Lasserre's method by adding the inequality 𝑐 -𝑓 ≥ 0 to the constraints 𝑔 1 ≥ 0, . . . , 𝑔 𝑚 ≥ 0. Remark 4.6. In [42, theorem 6.5], Lasserre states some classical conditions (known as the Karush-Kuhn-Tucker (KKT) conditions [42, section 7.1]), already encountered in nonlinear programming, to ensure the finite convergence of the hierarchy of the semidefinite relaxations (6). These conditions are initially a certificate for global optimality [42, theorem 7.4 and 7.5] and hold generically for a polynomial optimization problem [42, theorem 7.6].

Distance to the transversely isotropic stratum of the symmetric second-order tensor

Let V = S 2 (R 3 ) be the vector space of symmetric second-order tensors, endowed with the natural action 𝜌 2 (𝑔)a = 𝑔 a 𝑔 𝑇 , for a ∈ S 2 (R 3 ), 𝑔 ∈ 𝐺 = SO(3). Let q = (𝛿 𝑖𝑗 ) be the Euclidean metric,

a ′ = a - 1 3 tr(a) q,
be the traceless part of a, 𝜀 𝜀 𝜀 be the Levi-Civita tensor. We denote by ( ) 𝑠 the total symmetrization of a tensor. The generalized cross-product of symmetric tensors is defined as [61] (7)

a × b := -(a • 𝜀 𝜀 𝜀 • b) 𝑠 , (𝑖.𝑒., (a × b) 𝑖𝑗𝑘 := -(𝑎 𝑖𝑙 𝜀 𝑙𝑗𝑠 𝑏 𝑠𝑘 ) 𝑠 ) .
For a, b ∈ S 2 (R 3 ), it is a totally symmetric third-order tensor with 10 independent components. There are three isotropy classes for the symmetric second-order tensors (S 2 (R 3 ), SO(3)):

• [D 2 ] (orthotropy), if a has three distinct eigenvalues,

• [O( 2)] (transverse isotropy, characterized by the polynomial equation a 2 × a = 0 [61, Lemma 8.1]), if a has two distinct eigenvalues. • [SO(3)] (isotropy, characterized by the linear equation a ′ = 0), if a has three equal eigenvalues. We illustrate through this first example the accuracy of Lasserre's polynomial optimization method to compute the distance to an isotropy stratum. We shall obtain by this way the distance ∆(a 0 , Σ [O(2)] ) of the orthotropic second-order tensor

a 0 = ⎛ ⎝ -7 4 -4 4 5 -2 -4 -2 5 ⎞ ⎠
and compare the numerical results obtained with the algebraic solution derived in [4]: where 𝐾 = { a; a 2 × a = 0 } . Then, in order to properly apply the algorithm described in section 4, we need to ensure the Archimedean property (definition 3.2), and for that we use lemma 4.5. Therefore, to the 10 equations a 2 × a = 0, we add the inequality 𝑐 -‖a 0 -b‖ 2 ≥ 0 where we take

(8) ∆(a 0 , Σ [O(2)] ) 2 = ‖a 0 -a ** ‖ 2 = 18, where (9)
b = ⎛ ⎝ 1 0 0 0 1 0 0 0 -2 ⎞ ⎠ ∈ 𝐾,
and choose accordingly 𝑐 = 300. GloptiPoly then computes the approximation

(10) ‖a 0 -a * ‖ 2 of the minimum ∆(a 0 , Σ [O(2)] ) 2 = min a∈ ̃ 𝐾 ‖a 0 -a‖ 2 , where (11) ̃ 𝐾 = { a; a 2 × a = 0, 𝑐 -‖a 0 -a‖ 2 ≥ 0 } .
The optimal result computed in 1.2 seconds on a standard PC, is close to the exact solution ( 8)-( 9), with the constraints accurately satisfied:

∆(a 0 , Σ [O(2)] ) 2 ≈ 18.000007, a ** ≈ a * = ⎛ ⎝ -7.
max 𝑖 |𝑔 𝑖 (a * )| ‖a 0 ‖ 3 = max 𝑝,𝑞,𝑟 |(a * 2 × a * ) 𝑝𝑞𝑟 | ‖a 0 ‖ 3 = 5.696 10 -9 .
For different values 𝑐 ∈ [202, 450], one gets 18.000007 ≤ ∆(a 0 , Σ [O(2)] ) 2 ≤ 18.00006, with the GloptiPoly convergence obtained for the first degree of relaxation 𝑑 = 𝑑 0 = 2. The value chosen for 𝑐 affects the numerical solution. In fact, by increasing 𝑐 we get closer to the true minimum (=18), but the convergence is lost for 𝑐 ≥ 500 (with a GloptiPoly status 𝜉 = 0 at the first relaxation).

Remark 5.1. The transversely isotropic closed stratum Σ [O(2)] can also be characterized by a single scalar equation of degree 6, ( 12)

𝑔(a) = 12‖a 2 × a‖ 2 = ( tr(a ′ 2 ) ) 3 -6 ( tr(a ′3 ) ) 2 = 0,
with a ′ , the traceless part of a. However, there is no finite convergence of the associated relaxation problem, since grad a 𝑔(a) = 6𝑔(a) tr(a ′ 2 ) a ′ = 0 when 𝑔(a) = 0. In particular, the independence of the gradients of the constraint functions at the minimum (first order KKT sufficient condition mentioned in [42, theorem 6.5, theorem 7.2], see remark 4.6) is not satisfied.

The present example illustrates the strong dependence of the GloptiPoly convergence issue on the characterization of the isotropy classes. Indeed, convergence is obtained for the covariant characterization a 2 × a = 0, but not for the invariant characterization (12).

Distance to cubic elasticity isotropy stratum

In this section, we compute the distance of an experimental elasticity tensor E 0 to the cubic isotropy stratum Σ [O] , and determine the associated minimizer E * . The distance to an isotropy stratum problem has been widely addressed in the Continuum Mechanics literature, by solving it in terms of an unknown rotation (either parameterized by Euler angles [24,25], or by a unit quaternion [18,38,39]). Here, we use the characterization of the (cubic) isotropy stratum by means of at most quadratic covariants in order to formulate such a distance problem as a quadratic polynomial optimization problem. This makes us able to apply Lasserre's method, and to show that using GloptiPoly allows to compute an accurate solution of this non trivial example.

Formulation of the distance problem as a polynomial optimization problem. Let

V = Ela = { E ∈ ⊗ 4 R 3 , 𝐸 𝑖𝑗𝑘𝑙 = 𝐸 𝑗𝑖𝑘𝑙 = 𝐸 𝑘𝑙𝑖𝑗 } (dim Ela = 21)
be the set of elasticity tensors E : S 2 (R 3 ) → S 2 (R 3 ), introduced in example 2.1, and 𝐺 = SO(3). An elasticity tensor E ∈ Ela can be represented by a 6 × 6 symmetric matrix, in Voigt notation,

[E] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ (13)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
The vector space Ela decomposes into a direct sum of SO(3)-irreducible subspaces (so-called harmonic decomposition [7,68])

Ela = H 0 (R 3 ) ⊕ H 0 (R 3 ) ⊕ H 2 (R 3 ) ⊕ H 2 (R 3 ) ⊕ H 4 (R 3 ),
where H 𝑛 (R 3 ) denotes the space of harmonic tensors of order 𝑛 (dim H 𝑛 (R 3 ) = 2𝑛 + 1). Letting

d = tr 12 E, v = tr 13 E,
the harmonic decomposition of E ∈ Ela can be expressed as (see Appendix A for explicit formulas)

E = (𝛼, 𝛽, d ′ , v ′ , H), with 𝛼 = tr d, 𝛽 = tr v ∈ H 0 (R 3 ) the scalar (isotropic) components of E, with d ′ , v ′ ∈ H 2 (R 3 )
its second-order harmonic components (the traceless parts of d and v), and H ∈ H 4 (R 3 ) its fourth-order harmonic component. The squared Euclidean norm of E is then ( 14)

‖E‖ 2 = 5𝛼 2 + 4𝛽 2 + 2 21 ‖d ′ + 2v ′ ‖ 2 + 4 3 ‖d ′ -v ′ ‖ 2 + ‖H‖ 2 .
We consider the triclinic experimental elasticity tensor E 0 representing the Nickel-based aeronautics single crystal superalloy (of CMSX-4 type), measured in [25]. In Voigt notation: (15) [

E 0 ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 243
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ GPa.
This material has an expected symmetry, namely the cubic symmetry [O], deduced from its cubic microstructure (see figure 1). We then aim a computing of ( 16)

∆(E 0 , Σ [O] ) 2 = min E∈Σ O ‖E 0 -E‖ 2 .
This optimization problem has 21 variables 𝐸 𝑖𝑗𝑘𝑙 . To set it as a polynomial optimization problem, we take advantage of the fact that the cubic elasticity stratum is an algebraic set, characterized by explicit polynomial equations.

Theorem 6.1 (Olive et al [61]). Let E = (𝛼, 𝛽, d ′ , v ′ , H) ∈ Ela be an elasticity tensor,

d 2 = H . . . H (i.e., (d 2 ) 𝑖𝑗 = 𝐻 𝑖𝑝𝑞𝑟 𝐻 𝑝𝑞𝑟𝑗 ) and d ′ 2 = d 2 -1 3 tr(d 2 ) q be second-order covariants of E. Then E ∈ Σ [O] (is at least cubic) if and only if v ′ = d ′ = 0 and d ′ 2 = 0, and E ∈ Σ [O] (is cubic) if and only if furthermore H ̸ = 0.
We have then the following result.

Theorem 6.2. Let E = (𝛼, 𝛽, d ′ , v ′ , H) and E 0 = (𝛼 0 , 𝛽 0 , d ′ 0 , v ′ 0 , H 0 )
be two elasticity tensors. The 21-dimensional minimization problem (16) is equivalent to the 9-dimensional polynomial optimization problem min

d ′ 2 =0
‖H 0 -H‖ 2 , with E = (𝛼 0 , 𝛽 0 , 0, 0, H), and

O ‖E 0 -E‖ 2 = 2 21 ‖d ′ 0 + 2v ′ 0 ‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 ‖ 2 + min d ′ 2 =0 ‖H 0 -H‖ 2 . (17) min E∈Σ
Proof. The squared distance function in ( 16) is

‖E 0 -E‖ 2 = ‖(𝛼 0 -𝛼, 𝛽 0 -𝛽, d ′ 0 -d ′ , v ′ 0 -v ′ , H 0 -H)‖ 2 .
It can be expressed as

‖E 0 -E‖ 2 = 5(𝛼 0 -𝛼) 2 +4(𝛽 0 -𝛽) 2 + 2 21 ‖d ′ 0 +2v ′ 0 -(d ′ +2v ′ )‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 -(d ′ -v ′ )‖ 2 +‖H 0 -H‖ 2 ,
by using (14). By theorem 6.1, taking 𝛼 = 𝛼 0 , 𝛽 = 𝛽 0 , d ′ = 0 and v ′ = 0, we get (17), and the polynomial optimization problem ( 16) is reduced to the following problem in only 9 variables (the components of

H ∈ H 4 (R 3 ), dim H 4 (R 3 ) = 9) instead of 21, min d ′ 2 =0
‖H 0 -H‖ 2 . □ Remark 6.

3. An elasticity tensor E ∈ Ela corresponds to a quadratic elastic energy density, which must be positive semidefinite. This condition can be characterized using SO(3)-invariant polynomial inequalities on E, and thus added to the set of polynomial constraints, if necessary, using the following fact. Given a symmetric real 𝑛 × 𝑛 matrix 𝐴, we get

𝐴 is positive semidefinite ⇐⇒ ∀𝑖 ∈ {1, . . . , 𝑛} , 𝜎 𝑖 ≥ 0,
where 𝜎 1 , . . . , 𝜎 𝑛 denote the elementary symmetric polynomials in the eigenvalues 𝜆 𝑖 of 𝐴. Indeed, if 𝜆 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}, then 𝜎 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}. Conversely, assume that 𝜎 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}. Then, the polynomial

𝑝 := (𝑋 + 𝜆 1 )(𝑋 + 𝜆 2 ) . . . (𝑋 + 𝜆 𝑛 ) = 𝑋 𝑛 + 𝜎 1 𝑋 𝑛-1 + . . . + 𝜎 𝑛-1 𝑋 + 𝜎 𝑛 , satisfies 𝑝(𝑥) ≥ 𝑥 𝑛 > 0, ∀𝑥 > 0.
Hence, the (real) roots of 𝑝, namely -𝜆 1 , . . . , -𝜆 𝑛 , belong to ] -∞, 0], and thus 𝜆 𝑖 ≥ 0 for all 𝑖 ∈ {1, . . . , 𝑛}.

In practice, an experimental tensor E 0 is most often measured as semidefinite and the tensor the closest to E 0 computed as semidefinite, so, here we do not add the semidefiniteness constraint E ≥ 0 to our optimization problem.

6.2.

Resolution by Lasserre's method. A fourth order harmonic tensor H ∈ H 4 (R 3 ) is represented by the following real matrix (in Voigt notation) (18) [H] =

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
In practice we set

𝑥 𝑥 𝑥 = (Λ 1 , Λ 2 , Λ 3 , 𝑋 1 , 𝑋 2 , 𝑌 1 , 𝑌 2 , 𝑍 1 , 𝑍 2 )
, and GloptiPoly computes the approximation

(19) ‖H 0 -H * ‖ 2 of the minimum ∆(H 0 , Σ [O] ) 2 = min 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥), where 𝑓 (𝑥 𝑥 𝑥) = ‖H 0 -H‖ 2 and  𝐾 = { 𝑥 𝑥 𝑥; d ′ 2 = 0, 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0 } ,
with 𝑐 = 58000 > 𝑓 (0) to ensure the Archimedean property on the set of constraints. The five quadratic scalar equations (d ′ 2 ) 𝑖𝑗 = 0 are detailed in Appendix B. For E 0 given by ( 15), we have

𝑓 (𝑥 𝑥 𝑥) = 540Λ 2 + 620Λ 3 -88𝑋 1 + 668Λ 1 -264𝑍 1 -264𝑍 2 -456𝑋 2 + 8𝑍 2 2 + 8Λ 2 1 + 8Λ 2 2 + 8Λ 2 3 + 8𝑌 2 2 + 16𝑋 2 1 + 8𝑋 2 2 + 16𝑌 2 1 + 16𝑍 2 1 -392𝑌 1 -808𝑌 2 + 8𝑍 1 𝑍 2 + 2Λ 1 Λ 2 + 2Λ 2 Λ 3 + 8𝑋 1 𝑋 2 + 8𝑌 1 𝑌 2 + 2Λ 3 Λ 1 + 2026042 35 .
We obtain the result at the first relaxation order 𝑑 = 𝑑 0 = 1 with GloptiPoly status 𝜉 = +1 and value min

𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥) ≈ 𝑓 (𝑥 𝑥 𝑥 * ) = 2530.474727 GPa 2 ,
The computation time is of 0.9 seconds. The computed minimizer is By (18), it corresponds to the fourth-order harmonic tensor H * solution of (19). We get, by theorem 6.2, E * = (𝛼 0 , 𝛽 0 , 0, 0, H * ), i.e.,

𝑥 𝑥 𝑥 * = (-
E * = 1 15 (𝛼 0 + 2𝛽 0 ) q ⊙ q + 1 6 (𝛼 0 -𝛽 0 ) q ⊗ (2,2)
q + H * , with ⊙ the symmetric tensor product (see Appendix A). The elasticity tensor E * is cubic (and not isotropic) since H * ̸ = 0. Finally, the computed cubic tensor

E * ∈ Σ [O] the closest to E 0 is, in Voigt notation, [E * ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 240
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ GPa. It corresponds to ∆(E 0 , Σ [O] ) ≈ ‖E 0 -E * ‖ = 74.
131148 GPa and to the relative distance to cubic symmetry ‖E 0 -E * ‖ ‖E 0 ‖ = 0.103910, slightly better than the solution computed in [25] using a parameterization by Euler angles together with a simplex minimization method. Note that the constraint d ′ 2 = 0 is satisfied accurately, since

d ′ 2 ‖H 0 ‖ 2 = 10 -6 ⎛ ⎝
-4.097 -2.8 10 -6 -4.9 10 -6 -2.8 10 -6 -4.455 4.2 10 -6 -4.9 10 -6 4.2 10 -6 -8.552

⎞ ⎠ ≈ 0.
One can choose other values for 𝑐 satisfying 𝑐 > 𝑓 (𝑦 𝑦 𝑦) for some 𝑦 𝑦 𝑦 ∈ 𝐾. The GloptiPoly solution varies slightly as 𝑐 runs the interval [58000, 61000], with a computation time of 0.9 seconds for 𝑐 = 58000, of 0.8 seconds for 𝑐 = 60000, and of 0.1 seconds for 𝑐 = 61000. Outside from this narrow interval, the GloptiPoly convergence is lost (Gloptipoly status 𝜉 = 0). Remark 6.4. The computation time is lower for this quadratic optimization problem (with 9 variables) than for the degree 3 polynomial optimization problem of section 5 (with 6 variables).

Distance to cubic piezoelectricity isotropy stratum

In this final section, we apply Lasserre's polynomial optimization method to compute the distance ∆(e 0 , Σ [O -] ) of a raw piezoelectricity third-order tensor1 e 0 to the cubic piezoelectricity stratum Σ [O -] . This problem seems to have never been addressed before. It is important for the design of dielectric materials, since for instance the piezolectricity behavior strongly depends on the crystal primitive cell symmetry.

Formulation of the distance problem as a polynomial optimization problem.

According to the three-dimensional piezoelectricity framework [21,28], we denote by

V = Piez = { e ∈ ⊗ 3 R 3 , e 𝑖𝑗𝑘 = e 𝑖𝑘𝑗 } (dim Piez = 18),
the vector space of piezoelectricity tensors e : S 2 (R 3 ) → R The vector space Piez decomposes into a direct sum of O(3)-irreducible subspaces (so-called harmonic decomposition [68])

Piez = H 1 (R 3 ) ⊕ H 1 (R 3 ) ⊕ H 2♯ (R 3 ) ⊕ H 3 (R 3 ).
The notation H 𝑛 (R 3 ) still refers to the vector space of 𝑛-th order harmonic tensors endowed with the standard O(3)-representation 𝜌 𝑛 , while H 𝑛♯ (R 3 ) refers to the same vector space endowed with the twisted O(3)-representation ρ𝑛 , such that ρ𝑛 (𝑔) = (det 𝑔) 𝜌 𝑛 (𝑔). One has e = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a, h)

with 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 ∈ H 1 (R 3 ), a ∈ H 2♯ (R 3 ) and h ∈ H 3 (R 3 ).
Let ⊙ be the symmetric tensor product and e 𝑠 ∈ S 3 (R 3 ) denote the totally symmetric part of e (of components (e 𝑠 ) 𝑖𝑗𝑘 = 1 3 (e 𝑖𝑗𝑘 + e 𝑗𝑖𝑘 + e 𝑘𝑗𝑖 )). Any piezoelectricity tensor e ∈ Piez can be decomposed as the sum e = g + h where (20) h := e 𝑠 -3 5 q ⊙ tr(e 𝑠 ) ∈ H 3 (R 3 ), is the leading harmonic part of e, and g := eh = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a), is orthogonal to h (i.e., ⟨g, h⟩ = 𝑔 𝑖𝑗𝑘 ℎ 𝑖𝑗𝑘 = 0).

Remark 7.1. The third-order tensors g = g(e) and h = h(e) are linear covariants of e.

The squared Euclidean norm of e is then (21) ‖e‖ 2 = e 𝑖𝑗𝑘 e 𝑖𝑗𝑘 = ‖g‖ 2 + ‖h‖ 2 .

We will first consider the following raw (triclinic) piezoelectricity tensor e 0 for pure wurtzite AlN (aluminum nitride, 𝑥 = 0), of Voigt representation, (22) [e 0 ] = ⎛ ⎝ 0 0 -0.0505 -0.0394 -0.2854 -0.0637 -0.0637 -0.0042 0.0332 -0.2818 -0.0058 0.0185 -0.5807 -0.5822 1.4607 0.0022 0.0002 0.0043

⎞ ⎠ C/m 2 ,
in Coulomb per square meter, computed by Density Functional Theory (DFT), using abinitio simulations, by Manna and coworkers [49, Fig. 3]. We will also consider wurtzite alloys Cr 𝑥 Al 1-𝑥 N and the associated raw piezoelectricity tensors e 𝑥 0 (given in the Appendix D for chromium concentrations 0 ≤ 𝑥 ≤ 0.25). Note that pure rocksalt CrN corresponds to a Crconcentration 𝑥 = 1, and that the value 𝑥 = 0.25 is the so-called wurzite to rocksalt phase transition point [52].

We aim at computing by polynomial optimization (23) ∆(e 0 , Σ With the same proof as for theorem 6.2, we have the following result.

Theorem 7.3. Let e = g + h and e 0 = g 0 + h 0 be two piezoelectricity tensors, with h and h 0 their leading harmonic parts. The 15-dimensional minimization problem (23) is equivalent to the 7-dimensional polynomial optimization problem

min d ′ 2 =0
‖h 0 -h‖ 2 , with e = h, and min

e∈Σ O - ‖e 0 -e‖ 2 = ‖g 0 ‖ 2 + min d ′ 2 =0
‖h 0 -h‖ 2 .

7.2.

Resolution by Lasserre's method. A third order harmonic tensor h ∈ H 3 (R 1. Results for the raw piezoelectricity tensors of [49] for different Crconcentrations 𝑥 (the distance ∆(E 0 ,

Σ [O -] ) ≈ ‖e 0 -e * ‖ is in C/m 2 ).
Remark 7.5. The computation times are of the same order of magnitude as for the cubic elasticity case. For this quadratic optimization problem (with 7 variables) as well, they are lower than for the 6 variables but degree 3 optimization problem of section 5.

Conclusion

Some isotropy strata of tensorial representations of the orthogonal group are explicitly characterized by polynomial covariants. We have taken advantage of this fact to formulate the computation of the distance to these strata as a polynomial optimization problem. We have used the property that the isotropy classes for the representation of SO(3) on the vector space of elasticity tensors are in general semialgebraic. The present work shows the interest of the characterization of the isotropy classes by means of polynomial covariants, rather than by means of invariants. In particular, the covariant characterization of the cubic piezoelectricity symmetry stratum (theorem 7.2), which is the cornerstone of our methodology, is a new result.

We have then recalled Lasserre's method to solve polynomial optimization problems under semialgebraic constraints. Under the so-called Archimedean hypothesis, this approach consists in writing the initial problem as an infinite semidefinite program from which is constructed a sequence of relaxed semidefinite programs that converges to the desired global minimum. We have presented the corresponding algorithm implemented in the freeware GloptiPoly, in particular its stopping criterion.

We have applied this polynomial optimization method to compute the cubic tensor the closest to a raw (measured) constitutive tensor, both in continuum mechanics elasticity and piezoelectricity. We have considered the following examples

• of an elasticity tensor measured by François and coworkers [25] for an aeronautics Nickelbased single crystal superalloy, • of nine piezoelectricity tensors computed for wurtzite alloys using Density Functional Theory (DFT) and ab-initio simulations, by Manna and coworkers [49].

In both cases, we took advantage of the distance being a coercive polynomial function to adapt the constraints so that they can satisfy the Archimedean condition, in order to ensure the convergence of the method to the desired minimum.

Appendix A. Explicit harmonic decomposition of an elasticity tensor An elasticity tensor E ∈ Ela admits the following explicit harmonic decomposition [7]:

(26) E = E 𝑖𝑠𝑜 + q ⊗ (4) a + q ⊗ (2,2) b + H.
where ( 27)

E 𝑖𝑠𝑜 = 1 15 (𝛼 + 2𝛽) q ⊗ (4) q + 1 6 (𝛼 -𝛽) q ⊗ (2,2) q, (28)
a = 2 7 (d ′ + 2v ′ ), b = 2(d ′ -v ′ ). (29)
with d ′ = d -1 3 tr(d)q and v ′ = v -1 3 tr(v)q respectively the traceless parts of d = tr 12 E and v = tr 13 E.

In (26), q is the Euclidean canonical bilinear 2-form represented by the components (𝛿 𝑖𝑗 ) in any orthonormal basis and the tensor products ⊗ (4) and ⊗ (2,2) , between symmetric second-order tensors a, b, are defined as follows:

(a ⊗ We have, for Euclidean norm,

‖E‖ 2 = 5𝛼 2 + 4𝛽 2 + 2 21 ‖d ′ + 2v ′ ‖ 2 + 4 3 ‖d ′ -v ′ ‖ 2 + ‖H‖ 2 ,
Using ( 28)-( 29) we obtain for the experimental elasticity tensor E 0 (given by ( 15)) the harmonic decomposition

E 0 = (𝛼 0 , 𝛽 0 , d ′ 0 , v ′ 0 , H 0 ) with 𝛼 0 = 1531, 𝛽 0 = 1479, d ′ 0 = ⎛ ⎝ 11 3 2 14 2 5 3 23 14 23 -16 3 ⎞ ⎠ GPa , v ′ 0 = ⎛ ⎝ -1 -11 -1 -11 9 -1 -1 -1 -8 ⎞ ⎠ GPa, E 𝑖𝑠𝑜 0 = 1 15 (𝛼 0 + 2𝛽 0 ) q ⊗ (4) q + 1 6 (𝛼 0 -𝛽 0 ) q ⊗ (2,2) q,
and, in Voigt notation,

[H 0 ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -1986 35 1093 35 893 35 1093 35 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ GPa.
Appendix B. Components of second-order covariant d ′

Here, we will formulate alternative integrity bases for both R[H 3 (R 3 )] SO (3) and R[H 3 (R 3 )] O (3) , which happen to be more useful in order to characterize the cubic symmetry class in H 3 (R 3 ) for O(3). These will be used to prove theorem 7.2.

Theorem C.1. Let h ∈ H 3 (R 3 ) be an harmonic third-order tensor, d 2 = h : h, and

𝑣 𝑣 𝑣 3 := h : d ′ 2 , 𝑣 𝑣 𝑣 5 := d ′ 2 • 𝑣 𝑣 𝑣 3 , 𝑣 𝑣 𝑣 7 := d ′ 2 • 𝑣 𝑣 𝑣 5 where d ′ 2 = d 2 - 1 3 tr(d 2 )q.
(1) A minimal integrity basis of R[H 3 (R 3 )] SO (3) is constituted by the five invariants

𝐼 2 := tr d 2 = ‖h‖ 2 , 𝐼 4 := tr(d ′ 2 2 ) = ‖d ′ 2 ‖
(R 3 ) ⊕ H 1 (R 3 ) ⊕ H 2♯ (R 3 ), h ∈ H 3 (R 3 ). If e ∈ Σ [O -] , then g = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a) ∈ Σ [O -]
vanishes since an element in H 1 (R 3 ) or H 2♯ (R 3 ) with at least cubic symmetry ([O -]) is necessarily isotropic. For the same reason d ′ 2 (h) = 0. Conversely, if g = 0, then e = h ∈ H 3 (R 3 ) is harmonic, and it suffices to show that h ∈ Σ [O -] (is at least cubic). Since we assume furthermore d ′ 2 = 0, we have

𝐼 2 (h) = ‖h‖ 2 ≥ 0, 𝐼 4 (h) = 0, 𝐼 6 (h) = 0, 𝐼 10 (h) = 0.
Now an harmonic tensor in H 3 (R 3 ) which is fixed by O -is written (in Voigt notation ( 24)) as

h 0 = 𝛿 ⎛ ⎝ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ⎞ ⎠ .
For such a tensor we get

𝐼 2 (h 0 ) = 6𝛿 2 , 𝐼 4 (h 0 ) = 0, 𝐼 6 (h 0 ) = 0, 𝐼 10 (h 0 ) = 0.
Therefore, since 𝐼 2 (h) = ‖h‖ 2 ≥ 0, we can find a real number 𝛿 such that 6𝛿 (Perla Azzi) CNRS, Sorbonne université, IMJ -Institut de mathématiques Jussieu, 75005, Paris, France & Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS -Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France

Email address, Perla Azzi: perla.azzi@ens-paris-saclay.fr (Rodrigue Desmorat) Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.

Email address: rodrigue.desmorat@ens-paris-saclay.fr

Chapter 6

Distance to isotropy strata by Euler

Lagrange method

This chapter is an ArXiv preprint [4] in which we solve the same problem of the distance to isotropy strata introduced in the previous chapter but here using an analytic method for solving polynomial optimization problems based on Euler-Lagrange equations. Euler-Lagrange method for solving constrained polynomial optimization problems, consists in formulating the constrained problem to an unconstrained one with the help of Lagrange multipliers. This comes down to find the critical points of a polynomial function leading to a system of polynomial equations that can be solved using Gröbner basis (for example). With this method, we were able to solve the problem for some numerical examples, first in cubic elasticity and then in cubic elasto-plasticity for instance. However, a submersion condition is imposed on the constraints functions in order to apply this method which restrict the possible cases for which this method can be applied. For instance, finding the closest cubic piezoelectricity tensor to an experimental one, which has been already done for some examples using Lasserre's method in section 7 of the previous chapter, cannot be done using Euler-Lagrange method since the Jacobian of the constraint function does not have maximal rank and so the submersion condition is not satisfied.

The rotation 𝑄 can be parameterized by the Euler angles or by a unit quaternion q [21,39,40], allowing, in the second case, for the formulation of the considered distance problem as a polynomial optimization problem. Indeed, the function ‖E 0 -𝑄(q) ⋆ A‖ 2 is then quadratic in A and polynomial of degree 16 in q. Note, however, that a pair (𝑄, A) is far from representing uniquely a tensor E. For instance, a cubic elasticity tensor E is represented by 24 pairs (𝑄, A) or 48 pairs (q, A) (since 𝑄(q) = 𝑄(-q)). This means that we expect to find at least 24 global minima (𝑄, A), or 48 global minima (q, A) to problem (1), which correspond to the same cubic elasticity tensor E. More generally, there are at least as many global minima (𝑄, A) as there are symmetries of A (24 for cubic symmetry).

Since the variables 𝑄 and A are independent, if 𝐺 is the symmetry group of the considered material, the minimum (1) can be rewritten as [29] (2) min

𝑄 ‖E 0 -𝑄 ⋆ R 𝐺 (𝑄 𝑇 ⋆ E 0 )‖ 2 ,
where R 𝐺 is the Reynolds operators [60, Chapter 2] (orthogonal projector of Ela onto the vector space Fix 𝐺 of considered normal forms A) and, here for finite groups, defined explicitly by

R 𝐺 (C) := 1 |𝐺| ∑ 𝑅∈𝐺 𝑅 ⋆ C,
where |𝐺| is the order of the group 𝐺. Definition (2) is the expression of the distance to a symmetry class adopted by most authors. It has the advantage of removing one variable A in the minimization problem but the disadvantage of doubling the degree in 𝑄 and thus in 𝑞.

Solving the distance to a given symmetry problem ( 1) is usually done numerically with, then, the risk to reach a local minimum instead of a global one [29,47,20]. To overcome this difficulty, François and coworkers proposed to plot first pole figures for the given elasticity tensor E 0 [27,28] (renamed plots of the monoclinic distance in [47,40]). Accordingly, they got an initial value for E, not too far from E 0 , which was then optimized by a standard numerical (iterative) scheme.

Computational algebraic or semialgebraic optimization methods have been developed to find (directly) the global minimum of a multi-variable polynomial function (with polynomial constraints), using semidefinite programming for example [50,5,62,61,44,65,43]. On the other hand, there are nowadays symbolic computation methods to solve sets of polynomial equations, for instance the method of Gröbner bases [17] (see also [19,60]). These methods work well when the number of variables (i.e., of unknowns) is small and when the degree of the polynomials remains low [43]. The Gröbner basis method is available in the algebraic geometry software Macaulay2 [32] and in most Computer Algebra Systems. It does not make any numerical approximation if the coefficients of the considered polynomials are rational numbers and the Gröbner basis method can be seen as quasi-analytical. We use the prefix quasi because at one step, after an exact variables elimination process, one has to solve a polynomial equation in one variable, the remaining equations becoming afterwards linear.

The elasticity symmetry classes have been characterized by polynomial equations and inequalities in [53] (see also [2], or [9] for the case of harmonic fourth-order tensors), illustrating the mathematical property that the closed SO(3)-symmetry strata1 are semialgebraic sets [3,4,55,57]. The necessary and sufficient conditions for a Hooke tensor to belong to one of the eight elasticity symmetry strata have been formulated using polynomial covariants (in a coordinate-free manner). For a Hooke tensor, the cubic stratum is characterized by quadratic equations [53,Theorem 10.3]. Therefore, one hopes to formulate the distance to cubic elasticity problem as a quadratic optimization problem (of much lower degree than for the normal form/quaternion parameterization method) and expects a quasi-analytical solution using the Gröbner basis method. To succeed, one will simply have to derive first-order Euler-Lagrange equations for the corresponding quadratic optimization problem.

Cubic symmetry is of most importance for Ni-based single crystal superalloys, such as CMSX-4 [30,54,56], the material of aircrafts gas turbine blades (subject to (visco-)plasticity [45,13]). Thanks to the harmonic decomposition [10,58,18,11], the geometry of cubic fourth order tensors is now well understood. This will make it possible to formulate the calculation of the distance to cubic symmetry as a polynomial optimization problem, not only for a single elasticity tensor, but also for a pair (E, P) of two fourth-order constitutive tensors. Here, E is understood as the Hooke (elasticity) tensor and P as the Hill (plasticity) tensor. Indeed, we shall see that this pair of tensors is at least cubic if and only if the harmonic second-order components of E and P vanish and if their harmonic fourth-order components are at least cubic and proportional.

We will make use of the reformulation of the distance to cubic elasticity as a quadratic optimization problem, in order to solve it quasi-analytically. In practice, this will be done thanks to the theory of Gröbner bases. We will take advantage of the fact that the material parameters, such as the components of an experimental elasticity tensor, are measured with only a few significant digits to work with rational coefficients polynomials. This point is of main importance in the resolution of a system of polynomial equations by the obtention of a Gröbner basis (see remark A.1 in the Appendix).

The paper is organized as follows. The Euler-Lagrange method for solving constrained optimization problems is briefly presented in section 2. Background materials on cubic constitutive tensors are recalled in section 3 and section 4. The problem of the distance to cubic elasticity is formulated as a polynomial (quadratic) optimization problem in section 5 and solved thanks to the theory of Gröbner bases in section 6. The extension to the pair (E, P) of the Hooke and Hill tensors is described in section 7 and section 8. Finally, in section 9, we explain how to compute a natural cubic basis for any given cubic Hooke tensor. To be self-contained, a summary of Gröbner bases methods for algebraic elimination is provided in Appendix A. Last, the lack of accuracy of some upper bounds estimates of the distance to cubic elasticity is discussed in Appendix B.

Notations. We are working in orthonormal bases, so that we do not have to distinguish between covariant and contravariant tensors. The tensor product is denoted by ⊗. An harmonic tensor is a traceless totally symmetric tensor. The space of harmonic tensors of order 𝑛 will be denoted by H 𝑛 (R 3 ) or simply H 𝑛 . It is a subspace of dimension 2𝑛 + 1 of the vector space S 𝑛 (R 3 ), the space of totally symmetric tensors S = S 𝑠 of order 𝑛 (where (•) 𝑠 is the symmetrization operator).

Let S = S 𝑠 ∈ S 𝑝 (R 3 ) (of order 𝑝) and T = T 𝑠 ∈ S 𝑞 (R 3 ) (of order 𝑞) be two totally symmetric tensors. The totally symmetric tensor product ⊙ is defined by

S ⊙ T := (S ⊗ T) 𝑠 ∈ S 𝑝+𝑞 (R 3 ).
It is a totally symmetric tensor (of order 𝑝 + 𝑞). The generalized cross product between two totally symmetric tensors, which was introduced in [53], is defined by

(3) S × T := (T • 𝜀 𝜀 𝜀 • S) 𝑠 ∈ S 𝑝+𝑞-1 ,
where 𝜀 𝜀 𝜀 = (𝜀 𝑖𝑗𝑘 ) is the Levi-Civita tensor. In components, it is written as

(𝑇 𝑖 1 ...𝑖 𝑝-1 𝑘 𝜀 𝑘𝑖𝑝𝑙 𝑆 𝑙𝑖 𝑝+1 ...𝑖 𝑝+𝑞-1 ) 𝑠 .
A dot denotes a contraction between two tensors and several dots, several contractions. For instance

(a • b) 𝑖𝑗 = 𝑎 𝑖𝑘 𝑏 𝑘𝑗 , a : b = 𝑎 𝑖𝑗 𝑏 𝑖𝑗 , (H : a) 𝑖𝑗 = 𝐻 𝑖𝑗𝑘𝑙 𝑎 𝑘𝑙 , (H : K) 𝑖𝑗𝑘𝑙 = 𝐻 𝑖𝑗𝑝𝑞 𝐾 𝑝𝑞𝑘𝑙 , (H . . . K) 𝑖𝑗 = 𝐻 𝑖𝑝𝑞𝑟 𝐾 𝑝𝑞𝑟𝑗 ,
where a, b are second-order tensors and H, K, fourth-order tensors. The usual abbreviations H 2 = H : H and H 3 = H : H : H shall also be used.

The Euler-Lagrange method for polynomial functions and constraints

The simplest method to solve a minimization problem for a polynomial function 𝑓 , defined on R 𝑛 , is probably the Euler-Lagrange method, which consists in looking for its critical points. The critical points of 𝑓 are solutions of a system of algebraic equations which may be solved using Gröbner bases for example (see Appendix A). When, moreover, polynomial algebraic constraints 𝑔(𝑥 𝑥 𝑥) = 0 are involved, where 𝑔 : R 𝑛 → R 𝑝 , is a smooth vector-valued function, the method of Lagrange multipliers can be used [12,38,42]. In geometric terms, the constraint problem means that we seek for critical points of the restriction of 𝑓 to the submanifold of R 𝑛 This requires that the constraint function, 𝑔 : R 𝑛 → R 𝑝 , is a submersion on 𝑆 = 𝑔 -1 (0), which means that the linear tangent map (i.e., here the Jacobian matrix)

𝑇 𝑥 𝑥 𝑥 𝑔 : R 𝑛 → R 𝑝 is of maximal rank 𝑝 at each point 𝑥 𝑥 𝑥 ∈ 𝑆 (which requires that 𝑛 ≥ 𝑝). This condition ensures that 𝑆 is a smooth submanifold of R 𝑛 of dimension 𝑛 -𝑝 [42]. In that case, one can show, using the implicit function theorem, that the solutions of the constrained problem Remark 2.1. Note however that the extrema of 𝑓 are in general saddle points of 𝐹 [38]. Therefore, one should not make the false statement that the minimization of the constrained problem ( 5) is equivalent to the minimization of the function (6).

The critical points of ( 6) are the solutions of the algebraic system These equations are referred to as (first-order) Euler-Lagrange equations with constraints.

In practice, however, the problem is not that simple. In several problems, the set 𝑆 = 𝑔 -1 (0) contains some point 𝑥 𝑥 𝑥 at which 𝑔 is not a submersion. Worse, in the following example, which concerns the distance of a deviatoric second order tensors 𝑥 𝑥 𝑥 = h ∈ H 2 to transverse isotropy, the gradient of 𝑔 is singular at each point h of 𝑆.

Example 2.2 (The transversely isotropic (closed) strata in H 2 ). It is the vector subspace 𝑆 of H 2 of deviatoric tensors which have at least two identical eigenvalues. The set 𝑆 is defined implicitly by the polynomial equation

𝑔(h) := ( tr h 2 ) 3 -6 ( tr h 3 ) 2 = 0, h ∈ H 2 .
Unfortunately, the gradient of 𝑔 in H 2

grad h 𝑔 = 6 ( (tr h 2 ) 2 h -6(tr h 3 )(h 2 ) ′ )
vanishes identically on the set 𝑆 = 𝑔 -1 (0), since, when h ∈ H 2 is transversely isotropic, we have (h 2 ) ′ = tr h 3 tr h 2 h and thus

grad h 𝑔 = 6𝑔(h) tr h 2 h = 0.
Fortunately, all situations are not as bad as in this example but singularities may still exist. In the following examples, concerning respectively cubic fourth-order harmonic tensors and elasticity tensors, of main interest for the present work, the set 𝑆 is defined by a mapping 𝑔 which is a submersion on a big open subset of 𝑆, but not on all of 𝑆.

Example 2.3 (The cubic (closed) strata in H 4 ). It is the vector subspace 𝑆 of H 4 of fourth-order harmonic tensors which are at least cubic. It was shown in [53,Theorem 9.3], that this set can be defined as

𝑆 = { H ∈ H 4 ; 𝑔(H) = 0 } ,
where

𝑔 : H 4 → H 2
is a polynomial mapping of degree 2. One can check that 𝑔 is a submersion at each cubic tensor H ̸ = 0, but not at H = 0, which is a singular point.

Example 2.4 (The cubic (closed) strata in Ela). It is the vector subspace 𝑆 of Ela of elasticity tensors which are at least cubic. It was shown in [53,Theorem 10.2], that this set can be defined as

𝑆 = {E ∈ Ela; 𝑔(E) = 0} ,
where

𝑔 : Ela → H 2 ⊕ H 2 ⊕ H 2 is a polynomial mapping. One can check that 𝑔 is a submersion at each point E ∈ 𝑆 if E is cubic, but not if E is isotropic.
Remark 2.5. In [33] and [64] are proposed some algorithms to solve the constrained problem (5) even when the polynomial mapping 𝑔 is not a submersion, under some further hypotheses (the implementation of the algorithm proposed in [33] is available on the webpage of the first author as a Maple library). These algorithms involve the notions of nonsingular and singular points of the real algebraic set 𝑆 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔(𝑥 𝑥 𝑥) = 0}. Under some hypotheses on 𝑆 and on the polynomial coordinate functions (𝑔 1 , . . . , 𝑔 𝑝 ) of 𝑔, a point 𝑥 𝑥 𝑥 of 𝑆 is said to be nonsingular if the Jacobian matrix of 𝑔 at 𝑥 𝑥 𝑥 is of rank 𝑛 -𝑑, where 𝑑 is the dimension of the real algebraic set 𝑆 (which is by definition the so-called Krull dimension of the ring of polynomial functions on 𝑆), otherwise 𝑥 𝑥 𝑥 is said to be singular. If the real algebraic set 𝑆 has no singular point, it is said to be nonsingular and, in this case, 𝑆 is a smooth submanifold of R 𝑛 of dimension 𝑑 (precise definitions and properties can be found in [14]). However, the correctness of the subroutine GenCritValues of [64, Section 4] is, as far as we understand, not clear for us since it refers to an algorithm of [37] which is carried out on complex algebraic sets, not on real ones.

Cubic elasticity tensors

The space of elasticity tensors [26], denoted by Ela, is the space of fourth-order tensors E with the following index symmetries

𝐸 𝑖𝑗𝑘𝑙 = 𝐸 𝑗𝑖𝑘𝑙 = 𝐸 𝑖𝑗𝑙𝑘 = 𝐸 𝑘𝑙𝑖𝑗 .
Ela is a vector space of dimension 21 and an elasticity tensor E ∈ Ela can be represented in Voigt notation by the matrix

[E] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝐸 1111 = (1 -𝜈)𝐸 1 -𝜈 -2𝜈 2 , 𝐸 1122 = 𝜈𝐸 1 -𝜈 -2𝜈 2 , 𝐸 1212 = 𝐺,
and 𝐺 ̸ = 𝐸/2(1 + 𝜈) when E is cubic. In intrinsic notations, a cubic elasticity tensor E can be rewritten as in [26] (9)

E = 2𝜇 I + 𝜆1 ⊗ 1 + H, H ̸ = 0,
where I is the fourth order tensor with components 𝐼 𝑖𝑗𝑘𝑙 = A fourth-order harmonic tensor has 9 independent components. It can always be parameterized as (in Voigt notation, see [23,52]), (10) [

H] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
.

If H has at least the cubic symmetry, there exists an orthonormal basis (𝑒 𝑒 𝑒 𝑖 ), in which (in Voigt notation, see [9]):

(11) [H] = 𝛿 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 8 -4 -4 0 0 0 -4 8 -4 0 0 0 -4 -4 8 0 0 0 0 0 0 -4 0 0 0 0 0 0 -4 0 0 0 0 0 0 -4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 ) , 𝛿 = 1 4 (𝜇 -𝐺) ,
with 𝛿 = 0 when H is isotropic and 𝛿 ̸ = 0 when it is cubic.

Remark 3.1. The decomposition ( 9) of E into 𝜆, 𝜇, and H (with H cubic), is the so-called harmonic decomposition of a cubic elasticity tensor (see [10,18]).

The generalized Lamé constants 𝜆 = 𝜆(E), 𝜇 = 𝜇(E) are two (linear) invariants of E. The scalar 𝛿 = 𝛿(E) is a (rational) invariant of the cubic elasticity tensor E. Indeed, one has then [9, Section 5.1]:

(12) 𝛿 = 𝐽 3 4𝐽 2 , where (13)
𝐽 2 = ‖H‖ 2 = tr(tr 13 H 2 ) = 𝐻 𝑖𝑗𝑘𝑙 𝐻 𝑖𝑗𝑘𝑙 , and 𝐽 3 = tr(tr 13 H 3 ) = 𝐻 𝑖𝑗𝑘𝑙 𝐻 𝑘𝑙𝑝𝑞 𝐻 𝑝𝑞𝑖𝑗 , are two polynomial invariants of H (first introduced in [15]). The Euclidean squared norm of the cubic elasticity tensor E is then

‖E‖ 2 = 3 ( 3𝜆 2 + 4𝜆𝜇 + 8𝜇 2 ) + 480𝛿 2 .
When evaluated on (10), the invariants 𝐽 2 and 𝐽 3 can be expressed as

𝐽 2 = 2 ( 4Λ 2 1 + Λ 1 Λ 2 + Λ 1 Λ 3 + 4Λ 2 2 + Λ 2 Λ 3 + 4Λ 2 3 + 8𝑋 2 1 + 4𝑋 1 𝑋 2 + 4𝑋 2 2 (14) + 8𝑌 2 1 + 4𝑌 1 𝑌 2 + 4𝑌 2 2 + 8𝑍 2 1 + 4𝑍 1 𝑍 2 + 4𝑍 2 2 ) , 𝐽 3 = 6 ( Λ 2 1 Λ 2 + Λ 1 Λ 2 2 + Λ 2 3 (Λ 1 + Λ 2 ) -3𝑋 2 1 (Λ 1 + Λ 3 ) -2Λ 3 𝑋 1 𝑋 2 (15) + 4𝑋 1 (Λ 2 𝑋 2 + (𝑌 1 + 𝑌 2 )(𝑍 1 + 𝑍 2 )) + 𝑋 2 2 (Λ 2 + Λ 3 ) + 4𝑋 2 (𝑍 1 (𝑌 1 + 𝑌 2 ) + 𝑌 1 𝑍 2 ) -3Λ 1 𝑌 2 1 -3Λ 2 𝑌 2 1 + Λ 3 ( Λ 2 1 -Λ 1 Λ 2 + Λ 2 2 + 4𝑌 1 𝑌 2 + 𝑌 2 2 -3𝑍 2 1 ) -2Λ 1 𝑌 1 𝑌 2 + Λ 1 𝑌 2 2 -3Λ 2 𝑍 2 1 + 4Λ 1 𝑍 1 𝑍 2 -2Λ 2 𝑍 1 𝑍 2 + Λ 1 𝑍 2 2 + Λ 2 𝑍 2 2 ) . Remark 3.2.
Given a cubic elasticity tensor E * = (𝐸 * 𝑖𝑗𝑘𝑙 ), expressed in an arbitrary basis, the calculation of its normal form E = E O (of Voigt representation ( 8)) is straightforward (using ( 11) within ( 9)). Indeed, the normal form ( 8) is recovered from the calculation of 𝜆, 𝜇 and 𝛿 = 𝐽 3 /4𝐽 2 by the above formulas with

H = E 𝑠 -(2𝜇 + 𝜆)1 ⊙ 1 - 2 7 1 ⊙ (d ′ + 2v (17)

Cubic pair of elasticity-like tensors

There exist constitutive laws (for instance, anisotropic elasto-plasticity [35]) involving two fourth-order constitutive tensors. The question of the characterization of all the symmetry classes of a pair (E, F) ∈ Ela × Ela of elasticity-like tensors seems to be an open one. Nevertheless, this question has a relatively simple answer in the cubic symmetry case, thanks to the harmonic decompositions of both tensors E and F, E = (𝜆, 𝜇, d ′ , v ′ , H) and F = (ℓ, 𝑚, e ′ , w ′ , K), and by recalling that the symmetry group 𝐺 (E,F) of the pair (E, F) is the intersection of the symmetry groups of its harmonic components [26],

𝐺 (E,F) = 𝐺 E ∩ 𝐺 F = 𝐺 d ′ ∩ 𝐺 v ′ ∩ 𝐺 H ∩ 𝐺 e ′ ∩ 𝐺 w ′ ∩ 𝐺 K .
As an harmonic (deviatoric) cubic second-order tensor is isotropic and therefore vanishes (so that 𝐺 (E,F) = 𝐺 H ∩ 𝐺 K ), and as the normal form of an harmonic cubic fourth-order tensor is one-dimensional, the pair of elasticity-like fourth order tensors (E, F) is cubic if and only if its harmonic second-order components vanish and its harmonic fourth-order components are cubic and proportional. By theorem 3.3 we get the following result. To achieve this reduction, we perform the harmonic decompositions of both the given and the sought tensors E 0 and E (see remark 3.4 and ( 9)),

E 0 = (𝜆 0 , 𝜇 0 , d ′ 0 , v ′ 0 , H 0
) and E = (𝜆, 𝜇, 0, 0, H). Then, using the formula ( 18)

‖E‖ 2 = 3 ( 3𝜆 2 + 4𝜆𝜇 + 8𝜇 2 ) + 2 21 ‖d ′ + 2v ′ ‖ 2 + 4 3 ‖d ′ -v ′ ‖ 2 + ‖H‖ 2 ,
for the Euclidean squared norm ‖E‖ 2 := E :: E of an elasticity tensor E = (𝜆, 𝜇, d ′ , v ′ , H), we get

𝑓 (E) = ‖E 0 -E‖ 2 = 3 ( 3(𝜆 0 -𝜆) 2 + 4(𝜆 0 -𝜆)(𝜇 0 -𝜇) + 8(𝜇 0 -𝜇) 2 ) + 2 21 ‖d ′ 0 + 2v ′ 0 ‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 ‖ 2 + ‖H 0 -H‖ 2 ,
whose minimum for E = E * cubic is obtained for 𝜆 = 𝜆 0 , 𝜇 = 𝜇 0 and H cubic. We have therefore

E * = 2𝜇 0 I + 𝜆 0 1 ⊗ 1 + H * ,
with H * ∈ H 4 solution of the quadratic optimization problem ( 19) min

H ‖H 0 -H‖ 2 with 𝑔 = d ′ 2 = 0,
and the five scalar constraints in ( 5)

(d ′ 2 ) 11 = 0, (d ′ 2 ) 22 = 0, (d ′ 2 ) 12 = 0, (d ′ 2 ) 13 = 0, and (d ′ 2 ) 23 = 0, are indeed quadratic in H.
The optimum is cubic if H * ̸ = 0, with then the distance and the relative distance to cubic symmetry respectively equal to

𝑑(E 0 , cubic symmetry) = ‖E 0 -E * ‖ = √ 2 21 ‖d ′ 0 + 2v ′ 0 ‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 ‖ 2 + ‖H 0 -H * ‖ 2 ,
and 𝑑(E 0 , cubic symmetry)

‖E 0 ‖ = ‖E 0 -E * ‖ ‖E 0 ‖ .
In order to apply the Euler-Lagrange method to our constrained optimization problem (19), we have to check (see section 2) that the smooth mapping

𝑔 : H 4 → H 2 , H ↦ → d ′ 2 = (H . . . H) ′ ,
is a submersion for all cubic tensors H ∈ H 4 (i.e., that the Jacobian matrix 𝑇 H 𝑔 : H 4 → H 2 is of maximum rank 5, for each cubic tensor H). This is indeed the case. To show this, we observe that the mapping H ↦ → 𝑔(H) = (H . . . H) ′ is covariant, meaning that

𝑔(𝑄 ⋆ H) = 𝑄 ⋆ 𝑔(H)
for every rotation 𝑄. Therefore, the rank of 𝑇 H 𝑔 is equal to the rank of 𝑇 𝑄⋆H 𝑔 for every rotation 𝑄 and it is enough to compute this rank when H is the cubic normal form (11), which is 5. Note however that 𝑔 is not a submersion when H = 0 (i.e., when H is isotropic). The Euler-Lagrange method further reduces the distance problem (at given H 0 ), min

𝑔(H)=0 𝑓 (H), 𝑓 (H) = ‖H 0 -H‖ 2 , 𝑔(H) = d ′ 2 ,
to the determination of the critical points of the polynomial function ) ::

H = 𝜆 𝜆 𝜆 : d ′ 2 (H) = 0,
for each tensor H which satisfies d ′ 2 (H) = 0. Therefore, contracting four times the first equation H-H 0 +S(𝜆 𝜆 𝜆) = 0 in (23) with H, we obtain the second equation of ( 24), (H-H 0 ) :: H = 0.

Numerical application -Distance to cubic elasticity

Let us now apply the Euler-Lagrange method to the problem of determining the distance

𝑑(E 0 , cubic symmetry) = min E cubic ‖E 0 -E‖ = ‖E 0 -E * ‖,
of an experimental tensor E 0 to the cubic symmetry closed stratum. In our application the tensor E 0 , taken from [29] (refer to [41,8,7,28,21,16] for measurements), is the elasticity tensor of a Nickel-based single crystal superalloy. In Voigt notation: (26) [ It can be checked (by [53,Theorem 10.2 ], see also [29]) that the tensor E 0 is triclinic (with no material symmetry), even if it corresponds to a material with a so-called cubic 𝛾/𝛾 ′ microstructure [30,54,56].

E 0 ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 243
Using the formulas of theorem 3.3 we obtain the harmonic components of E 0 ,

𝜆 0 = 1583 15 GPa, 𝜇 0 = 1453 15 GPa, d ′ 0 = ⎛ ⎝ 11 3 2 14 2 5 3 23 14 23 -16 3 ⎞ ⎠ GPa, v ′ 0 = ⎛ ⎝ -1 -11 -1 -11 9 -1 -1 -1 -8 ⎞ ⎠ GPa
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ GPa.
The cost function 𝑓 = ‖H 0 -H‖ 2 to minimize can then be expressed as (in GPa 2 )

𝑓 (𝑥 𝑥 𝑥) =8Λ 2 1 + 2Λ 1 Λ 2 + 2Λ 1 Λ 3 + 668Λ 1 + 8Λ 2 2 + 2Λ 2 Λ 3 + 540Λ 2 + 8Λ 2 3 + 620Λ 3 + 16𝑋 2 1 + 8𝑋 1 (𝑋 2 -11) + 8𝑋 2 2 -456𝑋 2 + 16𝑌 2 1 + 8𝑌 1 𝑌 2 -392𝑌 1 + 8𝑌 2 2 -808𝑌 2 + 16𝑍 2 1 + 8𝑍 1 𝑍 2 -264𝑍 1 + 8𝑍 2 2 -264𝑍 2 + 2026042 35 in the variable 𝑥 𝑥 𝑥 = (𝑋 1 , 𝑋 2 , 𝑌 1 , 𝑌 2 , 𝑍 1 , 𝑍 2 , Λ 1 , Λ 2 , Λ 3 ). if the parameterization (10) is used for H.
In terms of components, and according to the expression (26) for the considered material, the system of equations of lemma 5.1 is constituted (a) of the three scalar equations,

𝑋 1 = 1515991Λ 1 + 6907074Λ 2 + 2816520Λ 3 + 4774213𝑌 2 + 1319317𝑍 1 + 3827136𝑍 2 2851559 , 𝑋 2 = -2752251Λ 1 -5474665Λ 2 -1823999Λ 3 -3665127𝑌 2 + 1198746𝑍 1 -1655027𝑍 2 2851559 , 𝑌 1 = -1401385Λ 1 -23691851Λ 2 -1939864Λ 3 -15828579𝑌 2 + 4529623𝑍 1 + 4531405𝑍 2 8554677 ,
which is here given by the solution 𝑠 = 1, Λ 3 = Λ

(1)

-E * ‖ ‖E 0 ‖ = 0.1039,
it is slightly better than the solution obtained by François-Geymonat-Berthaud by a numerical iterative method [29].

As H * ̸ = 0, the tensor E * is cubic. The distance of E 0 to isotropy,

𝑑(E 0 , isotropy) = ‖E 0 -(2𝜇 0 I + 𝜆 0 1 ⊗ 1)‖ = 246.68 GPa,
is found larger than the one to cubic symmetry, with a relative distance to isotropy 𝑑(E 0 , isotropy) ‖E 0 ‖ = 0.3458.

By remark 3.2, the normal form (denoted here by E * O ) of the optimal cubic elasticity tensor E * = (𝜆 = 𝜆 0 , 𝜇 = 𝜇 0 , 0, 0, H * ) given by (28), is obtained directly thanks to the computation of its invariants. We get, by the explicit formulas (12) GPa, which, for practical applications, can be by approximated by (30) [ GPa.

E * O ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 213

Distance to cubic elasto-plasticity as a polynomial optimization problem

The anisotropic Hill elasto-plasticity theory for metallic materials introduces not one but two fourth-order constitutive tensors [35,45,36,13],

• a first one, E ∈ Ela, to describe the anisotropic elasticity,

• a second one, P (sometimes considered as dimensionless), to describe the yield (plasticity) criterion, and such as the condition 𝜎 𝜎 𝜎 ′ : P : 𝜎 𝜎 𝜎 ′ -𝑅 2 < 0 corresponds to an elastic loading or unloading stage (with 𝜎 𝜎 𝜎 ′ ∈ H 2 the continuum mechanics deviatoric stress tensor). When assumed constant, the scalar 𝑅 stands for the material yield stress, when taken as evolving during loading, it stands for the material hardening. The Hill tensor P has the indicial symmetries of elasticity tensors (so that P ∈ Ela). With no lack of generality, instead of P, we can work with a tensor F of elasticity-type, and compute a dimensionless Hill tensor P = F/𝐶 by normalizing afterward F with a constant 𝐶. Indeed, when F is in its normal form (8) ) .

The harmonic decomposition of F is then (see remark 3.4) are respectively the dilatation and Voigt tensors of F, and K = (F) 𝑠′ ∈ H 4 is the harmonic fourth-order component of F, given by ( 17),

F = ( ℓ, 𝑚, e ′ , w ′ , K ) , with ℓ = 1
K = (F) 𝑠 -(2𝑚 + ℓ)1 ⊙ 1 - 2 7 1 ⊙ (e ′ + 2w ′ ).
We now assume that two given elasto-plasticity tensors E 0 and F 0 are available (possibly triclinic) for a given metallic material. As a generalization of the formulation of the distance problem of section 5, in which only one constitutive tensor (the elasticity tensor) was involved, we propose to define the optimum cubic estimates E ** and F ** = 𝐶P ** of the two elastoplasticity constitutive tensors, as the minimizers of the following quadratic function (with 𝑊 a given strictly positive weight)

𝑓 (E, F) := ‖E 0 -E‖ 2 + 𝑊 ‖F 0 -F‖ 2 ,
at given tensors E 0 and F 0 = 𝐶P 0 , under the constraint that both the elasticity tensor E and the Hill tensor F = 𝐶P are cubic and share the same cubic axes (by theorem 4.1). The introduction of a weight 𝑊 is necessary in practice, since the orders of magnitude (and the units) of the Hooke and Hill tensors are often very different.

Remark 7.1.

√ ‖E‖ 2 + 𝑊 ‖F‖ 2 , with 𝑊 > 0, is a norm on Ela ⊕ Ela.

We first perform the harmonic decompositions of E 0 and F 0 ,

E 0 = ( 𝜆 0 , 𝜇 0 , d ′ 0 , v ′ 0 , H 0 ) , F 0 = ( ℓ 0 , 𝑚 0 , e ′ 0 , w ′ 0 , K 0 ) , with 𝜆 0 , 𝜇 0 , ℓ 0 , 𝑚 0 ∈ H 0 , d ′ 0 , v ′ 0 , e ′ 0 , w ′ 0 ∈ H 2
and H 0 , K 0 ∈ H 4 their harmonic components. The harmonic decompositions of the sought cubic tensors E and F are E = (𝜆, 𝜇, 0, 0, H) , F = (ℓ, 𝑚, 0, 0, K) , with 𝜆, 𝜇, ℓ, 𝑚 ∈ H 0 , H, K ∈ H 4 , and, according to (9), we have

E = 2𝜇I + 𝜆1 ⊗ 1 + H and F = 2𝑚I + ℓ1 ⊗ 1 + K, with d ′ 2 (H) = d ′ 2 (
K) = 0 and K = 𝑘H (by theorem 3.3). Using the formula (18) for both ‖E 0 -E‖ 2 and ‖F 0 -F‖ 2 , we get

𝑓 (E) = 3 ( 3(𝜆 0 -𝜆) 2 + 4(𝜆 0 -𝜆)(𝜇 0 -𝜇) + 8(𝜇 0 -𝜇) 2 ) + 3𝑊 ( 3(ℓ 0 -ℓ) 2 + 4(ℓ 0 -ℓ)(𝑚 0 -𝑚) + 8(𝑚 0 -𝑚) 2 ) + 2 21 ‖d ′ 0 + 2v ′ 0 ‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 ‖ 2 + 2𝑊 21 ‖e ′ 0 + 2w ′ 0 ‖ 2 + 4𝑊 3 ‖e ′ 0 -w ′ 0 ‖ 2 + ‖H 0 -H‖ 2 + 𝑊 ‖K 0 -𝑘H‖ 2 .
The minimum of this expression is obtained for

𝜆 = 𝜆 0 , 𝜇 = 𝜇 0 , ℓ = ℓ 0 , 𝑚 = 𝑚 0 , H = H * , 𝑘 = 𝑘 * ,
where H * and 𝑘 * correspond to absolute minima of the problem min

H,𝑘 { ‖H 0 -H‖ 2 + 𝑊 ‖K 0 -𝑘H‖ 2 } , with d ′ 2 (H) = (H . . . H) ′ = 0.
Remark 7.2. Note that the condition K = 𝑘H ̸ = 0 implies that the pair (H, K) is cubic, meaning that both H and K are cubic and share the same cubic axes.

To solve the problem of the distance of a pair (E 0 , F 0 ) to cubic symmetry, we therefore have to find the critical points of the polynomial function (32) 𝐹 (H, 𝑘, 𝜆 𝜆 𝜆) := ‖H 0 -H‖ 2 + 𝑊 ‖K 0 -𝑘H‖ 2 + 𝜆 𝜆 𝜆 : 𝑔, with H ∈ H 4 an harmonic fourth-order tensor, 𝑘 a scalar, and where the Lagrange multiplier 𝜆 𝜆 𝜆 ∈ H 2 is a deviatoric second-order tensor. Observe that the first-order Euler-Lagrange equations for this optimization problem can furthermore be recast in a similar form as (24). The first equation of ( 33) is not linear anymore, it cannot be used to reduce the number of unknowns before the computation of a Gröbner basis. The quasi-analytical resolution by the obtention of a Gröbner basis will nevertheless be similar (but with four more variables) to the resolution for the single elasticity tensor case (except that the computation of a Gröbner basis will be more computer time consuming).

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ H . . . (H 0 + 𝑘𝑊 K 0 ) -(H 0 + 𝑘𝑊 K 0 ) . . . H = 0 (3 scalar equations) (H -H 0 ) :: H = 0 (1 scalar equation) 𝑊 (𝑘H -K 0 ) :: H = 0 (1 scalar equation) d ′ 2 = 0 (33)

Numerical application -Distance to cubic elasto-plasticity

We consider here the example of the triclinic elasticity tensor E 0 (still given by ( 26), the harmonic decomposition E 0 = (𝜆 0 , 𝜇 0 , d ′ 0 , w ′ , H 0 ) remaining the one of section 6), and of the following triclinic plasticity tensor F 0 , in Voigt notation, Using the formulas of theorem 3.3, we obtain F 0 = (ℓ 0 , 𝑚 0 , e ′ 0 , w ′ 0 , K 0 ), with

[F 0 ] = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 191
ℓ 0 = - 324 5 , 𝑚 0 = 1853 15 , e ′ 0 = (tr 12 F 0 ) ′ = ⎛ ⎝ 4 3 0 -4 0 -5 3 -7 -4 -7 1 3 ⎞ ⎠ , w ′ 0 = (tr 13 F 0 ) ′ = ⎛ ⎝ 59
( ‖E 0 -E‖ 2 + 𝑊 ‖F 0 -F‖ 2 ) , is (in GPa 2 ) 𝑓 (𝑥 𝑥 𝑥) = 8Λ 2 1 + 2Λ 1 Λ 2 + 2Λ 1 Λ 3 + 668Λ 1 + 8Λ 2 2 + 2Λ 2 Λ 3 + 540Λ 2 + 8Λ 2 3 + 620Λ 3 + 16𝑋 2 1 + 8𝑋 1 (𝑋 2 -11) + 8𝑋 2 2 -456𝑋 2 + 16𝑌 2 1 + 8𝑌 1 𝑌 2 -392𝑌 1 + 8𝑌 2 2 -808𝑌 2 + 16𝑍 2 1 + 8𝑍 1 𝑍 2 -264𝑍 1 + 8𝑍 2 2 -264𝑍 2 + 2026042 35 + 𝑊 ( 16𝑋 2 1 𝑘 2 + 8𝑋 1 𝑘(𝑋 2 𝑘 -70) + 8𝑋 2 2 𝑘 2 + 920𝑋 2 𝑘 + 16𝑌 2 1 𝑘 2 + 8𝑌 1 𝑌 2 𝑘 2 + 160𝑌 1 𝑘 + 8𝑌 2 2 𝑘 2 + 1792𝑌 2 𝑘 + 8Λ 2 1 𝑘 2 + 2Λ 1 Λ 2 𝑘 2 + 2Λ 1 Λ 3 𝑘 2 + 8Λ 2 2 𝑘 2 + 2Λ 2 Λ 3 𝑘 2 + 8Λ 2 3 𝑘 2 + 16𝑘 2 𝑍 2 1 + 8𝑘 2 𝑍 1 𝑍 2 + 8𝑘 2 𝑍 2 2 -258Λ 1 𝑘 + 304Λ 2 𝑘 -318Λ 3 𝑘 -344𝑘𝑍 1 + 656𝑘𝑍 2 + 6495682 35
) .

It is expressed in the variable

𝑥 𝑥 𝑥 = (𝑘, 𝑋 1 , 𝑋 2 , 𝑌 1 , 𝑌 2 , 𝑍 1 , 𝑍 2 , Λ 1 , Λ 2 , Λ 3 ),
if the parameterization (10) is used for H. The first-order Euler-Lagrange equations are given in lemma 7.3. In components, they consist of (a) the three scalar equations,

(𝑎1) -43𝑘Λ 1 𝑊 + 125𝑘Λ 2 𝑊 + 328𝑘Λ 3 𝑊 + 𝑋 1 (199 -264𝑘𝑊 ) + 𝑋 2 (52 -204𝑘𝑊 ) -555𝑘𝑊 𝑌 1 -70𝑘𝑊 𝑌 2 + 220𝑘𝑊 𝑍 1 + 159𝑘𝑊 𝑍 2 -33Λ 1 -132Λ 3 + 138𝑌 1 -11𝑌 2 -187𝑍 1 -310𝑍 2 = 0, (𝑎2) 𝑘𝑊 (-204Λ 1 -896Λ 2 -20Λ 3 + 375𝑋 1 + 43𝑋 2 + 61𝑌 1 + 152𝑌 2 -25𝑍 1 + 185𝑍 2 ) + 52Λ 1 + 404Λ 2 + 49Λ 3 + 33𝑋 2 + 123𝑌 1 + 270𝑌 2 -79𝑍 1 -46𝑍 2 = 0, (𝑎3) 460𝑘Λ 1 𝑊 -70𝑘Λ 2 𝑊 + 185𝑘Λ 3 𝑊 -7𝑋 1 (13𝑘𝑊 + 21) + 𝑋 2 (129𝑘𝑊 -334) + 4𝑘𝑊 𝑌 1 -125𝑘𝑊 𝑌 2 -612𝑘𝑊 𝑍 1 + 20𝑘𝑊 𝑍 2 -228Λ 1 -11Λ 2 -46Λ 3 + 99𝑌 1 + 156𝑍 1 -49𝑍 2 = 0, which correspond to the linear equation H . . . (H 0 + 𝑘𝑊 K 0 ) -(H 0 + 𝑘𝑊 K 0 ) . . . H = 0, (
(𝑘H -K 0 ) :: H = 8𝑘Λ 2 1 + 2𝑘Λ 1 Λ 2 + 2𝑘Λ 1 Λ 3 + 8𝑘Λ 2 2 + 2𝑘Λ 2 Λ 3 + 8𝑘Λ 2 3 + 16𝑘𝑋 2 1 + 8𝑋 1 (𝑘𝑋 2 -35) + 8𝑘𝑋 2 2 + 16𝑘𝑌 2 1 + 8𝑘𝑌 1 𝑌 2 + 8𝑘𝑌 2 2 + 16𝑘𝑍 2 1 + 8𝑘𝑍 1 𝑍 2 + 8𝑘𝑍 2 2 -129Λ 1 + 152Λ 2 -159Λ 3 + 460𝑋 2 + 80𝑌 1 + 896𝑌 2 -172𝑍 1 + 328𝑍 2 = 0,
(d) and the 5 equations 𝑔 𝑖𝑗 = (d 2 ) ′ 𝑖𝑗 = 0 (detailed in point (c) of section 6). We set a unit weight 𝑊 = 1 for the numerical application. The resolution is similar to the one for the single elasticity tensor case, except that now the variable 𝑥 𝑥 𝑥 is 10-dimensional, and that there is no a priori reduction in the number of scalar unknowns. Rational coefficients are considered for the given tensors E 0 and F 0 (and for their harmonic components H 0 and K 0 ). A Gröbner basis GB = {GB 1 , . . . , GB 111 } of 111 elements is computed using Mathematica. Its first element GB 1 is found to be a polynomial in Λ 3 only; Λ 3 is either zero (leading to the isotropic solution H = 0) or it is a solution of a polynomial equation of degree 56, which has 18 real non zero roots (in practice determined with a 100 significant digits precision). Once GB 1 (Λ 3 ) = 0 is solved, the remaining Gröbner basis equations are linear (as in (38) of Appendix A) in the variables Λ 2 , Λ 1 , . . . , 𝑋 2 , 𝑋 1 , and 𝑘.

The minimum minimorum for the cost function is here given by the solution Λ 3 = -19.612165 (it is not given by the isotropic solution H = K = 0). We get the optimal value 𝑘 ** = -2.134021 for 𝑘 and (in GPa): With the values 𝜆 = 𝜆 0 = 105.533333 and 𝜇 = 𝜇 0 = 96.866667, the optimal cubic elasticity tensor

𝑋 1 = -16.
E ** = 2𝜇 0 I + 𝜆 0 1 ⊗ 1 + H ** , has Voigt representation, [E ** ] = ⎛ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎠ GPa.
Since ℓ = ℓ 0 = -64.800000 and 𝑚 = 𝑚 0 = 123.533333, we get for the optimal cubic plasticity tensor The relative distance to cubic symmetry for this two constitutive elasto-plasticity tensors problem is

F ** = 2𝑚 0 I + ℓ 0 1 ⊗ 1 + 𝑘 *
√ ‖E 0 -E ** ‖ 2 + ‖F 0 -F ** ‖ 2 ‖E 0 ‖ 2 + ‖F 0 ‖ 2 = 0.2462.
It is slightly larger than the relative distance for the single elasticity tensor case solved in section 6.

As H ** and K ** = 𝑘 ** H ** are non zero, the two optimal tensors E ** and F ** are cubic (and so is the pair (E ** , F ** )). The relative distance of the given pair (E 0 , F 0 ) to isotropy,

√ ‖E 0 -(2𝜇 0 I + 𝜆 0 1 ⊗ 1)‖ 2 + ‖F 0 -(2𝑚 0 I + ℓ 0 1 ⊗ 1)‖ 2 ‖E 0 ‖ 2 + ‖F 0 ‖ 2 = 0.5096,
is larger than the one to cubic symmetry. The normal forms (8) for both the optimal Hooke and Hill tensors are finally obtained thanks to the computation of their invariants 𝜆, 𝜇 𝐽 2 , 𝐽 3 and 𝛿, here evaluated first for E ** and then for F ** (by remark 3.2). Using (16) for each tensor E ** and F ** , we get: GPa.

These two normal forms are obtained in the same cubic basis (𝑒 𝑒 𝑒 𝑖 ). Finally, by (31), the Hill parameter associated with F ** O is 𝐿 = 0.312949 ≈ 0.31.

9.

Recovering the natural basis of a cubic fourth-order constitutive tensor A continuum mechanics anisotropic constitutive law, such as elasticity, is not represented by a unique constitutive tensor E but by the set of all elasticity tensors 𝑄 ⋆ E related to E by a rotation 𝑄. Mathematically speaking, the anisotropic material property is represented by the orbit Orb(E) = {𝑄 ⋆ E, det 𝑄 = 1} .

For a given cubic elasticity tensor E, there exists a tensor E O in its orbit that is fixed by all the transformations of the orientation preserving octahedral group O. The tensor E O is the so-called normal form of E, and has (8) as Voigt representation. When a cubic constitutive tensor -such as the tensors E * , E ** and F ** of previous numerical applications sections -is not expressed in its natural (cubic) basis, one needs (1) to compute its normal form, (2) and to compute the rotation 𝑄 that puts it in its normal form.

Task (1) can be done in a straightforward manner, using Invariant Theory (see remark 3.2). Note that the polynomial (𝜆, 𝜇, 𝐽 2 and 𝐽 3 ) and rational (𝛿) invariants then involved are computed in the working basis (in which are expressed E * , E ** and F ** ) by explicit formulas, whereas the methodology proposed in [59] needs the computation of the eigenvalues of the Kelvin 6 × 6 matrix representation of the considered elasticity tensor.

In practice, there are several ways to perform task (2): using Maxwell multipoles [11] and solving a degree-8 polynomial equation in one variable, or solving the linear system [2, Appendix B]

L(a) := tr(H × a) = 0, a ∈ H 2 ,
where H is the fourth-order harmonic component of the considered (cubic) elasticity tensor E (it will next be either H * or H ** or K ** ). Here, the product × is the generalized cross product between totally symmetric tensors, defined by (3), and the totally symmetric fifth-order tensor H × a has components (H × a) 𝑖𝑗𝑘𝑙𝑚 = (𝑎 𝑖𝑟 𝜀 𝑟𝑗𝑠 𝐻 𝑠𝑘𝑙𝑚 ) 𝑠 .

Generically, the deviatoric tensor a, solution of the equation L(a) = 0, is orthotropic and carries the cubic basis (𝑒 𝑒 𝑒 𝑖 ). We shall apply the second methodology, which reduces to solve the linear equation L(a) = 0, once the components of a cubic elasticity tensor are given (in an arbitrarily oriented basis).

Remark 9.1. To avoid useless computations, it is important to note that, given a cubic elasticity tensor E = (𝜆, 𝜇, 0, 0, H), it is equivalent to solve tr(H × a) = 0, a ∈ H 2 , or to solve tr(E 𝑠 × a) = 0, a ∈ H 2 , where E 𝑠 is the totally symmetric part of E.

The leading harmonic part H of E is assumed to be known. Indeed, it has been computed in the previous applications sections for the three optimal tensors E * , E ** and F ** . The methodology to determine the rotation matrix 𝑄 is the following.

(1) Compute a basis (a 1 , a 2 ) of the two-dimensional space of solutions of the linear system L(a) = 0. (2) The pair of second-order tensors (a 1 , a 2 ) is orthotropic [53]. Hence, a random tensor a = 𝑡a 1 + 𝑠a 2 in this subspace will be generically orthotropic (as also, almost certainly, both a 1 , a 2 computed by a Computer Algebra System). For such an orthotropic tensor, an orthogonal basis of eigenvectors 𝑢 𝑢 𝑢 𝑖 will provide the solution as the rotation matrix 𝑄 = (𝑢 𝑢 𝑢 1 , 𝑢 𝑢 𝑢 2 , 𝑢 𝑢 𝑢 3 ). (3) The normal form E O of E is then obtained as

E O = 𝑄 ⋆ E, (E O ) 𝑖𝑗𝑘𝑙 = 𝑄 𝑖𝑝 𝑄 𝑗𝑞 𝑄 𝑘𝑟 𝑄 𝑙𝑠 𝐸 𝑝𝑞𝑟𝑠 .
Rotation associated with the cubic normal form for E * . Let us first apply this methodology to the cubic tensor E = E * = (𝜆 0 , 𝜇 0 , 0, 0, H = H * ) the nearest to E 0 (given by ( 28)). A basis for the space of traceless solutions for the system L(a) = tr(H * × a) = 0 is Rotation associated with the cubic normal forms for E ** and F ** . The methodology also applies to the optimal cubic tensors E ** and F ** of section 8 . A basis for the space of traceless solutions for the system tr(H ** × a) = 0 is

a 1 = ( 1
a 1 = ( 1

Conclusion

Thanks to the recent characterization of the cubic elasticity symmetry classes by polynomial covariants [53], we have formulated the distance to cubic symmetry problem as a polynomial optimization problem, and derived the associated Euler-Lagrange equations. We have used the theory of Gröbner bases to solve these equations, in a quasi-analytical manner (using a Computer Algebra System). This methodology has been applied to the case of a single elasticity tensor, as well as to the case of a pair of Hooke and Hill elasto-plasticity tensors. Besides, we have recovered the normal forms of the optimal cubic elasticity/plasticity tensors.

The key-point of the study is that the corresponding cubic symmetry is defined by a polynomial tensorial equation, which is a submersion (apart from the isotropic singularity, which is controlled). This makes it possible to apply the Euler-Lagrange method and use Gröbner bases to compute the critical points. Therefore, in this example, computing a Gröbner basis for a system of 𝑛 equations in 𝑛 variables leads to an equivalent system of equations of the form (38)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑃 𝑛 (𝑥 𝑛 ) = 0 𝑥 𝑛-1 = 𝑃 𝑛-1 (𝑥 𝑛 ) • • • 𝑥 1 = 𝑃 1 (𝑥 𝑛 )
In other words, in that case, one has been able to reduce the non-linear algebraic system (37) to an equivalent triangular system consisting in one polynomial equation in the last variable 𝑥 𝑛 and a list of 𝑛 -1 equations which are solved in the remaining variables 𝑥 𝑛-1 , . . . , 𝑥 1 . In particular, such a system has at most a finite number of solutions.

This is generally what happens if one tries to solve an algebraic system of 𝑛 equations in 𝑛 variables but there exists, nevertheless, some degenerate situations (as it is the case for linear systems when the determinant of the system vanishes). In the next example, we will illustrate this degeneracy. Consider the following set of equations ( 39)

⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 2 1 + 𝑥 2 2 + 𝑥 2 3 = 1 𝑥 1 𝑥 2 + 𝑥 2 3 = 1 𝑥 2
2 -𝑥 1 𝑥 2 + 𝑥 2 1 = 0. An equivalent system given by the computation of a Gröbner basis (for the lexicographic order induced by

𝑥 1 < 𝑥 2 < 𝑥 3 ) is ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑥 2 3 + 𝑥 1 𝑥 2 -1 = 0 -𝑥 2 𝑥 2 3 + 𝑥 1 𝑥 2 3 -𝑥 3 2 + 𝑥 2 -𝑥 1 = 0 𝑥 4 3 + 𝑥 2 2 𝑥 2 3 -2𝑥 2 3 + 𝑥 4 2 -𝑥 2 2 + 1 = 0 𝑥 2
3 + 𝑥 2 2 + 𝑥 2 1 -1 = 0 This time, the explicit solution of the problem is far less straightforward (it will be given and explained anyway below). What we can observe, however, is that the third equation in (39) is a linear combination of the first two ones. Thus, the system is in fact rectangular (two equations in three variables, rather than three equations). More generally, this situation appears for systems of 𝑛 algebraic equations with 𝑛 variables each time the polynomials 𝑓 1 , . . . , 𝑓 𝑛 are algebraically dependent.

Let us discuss now in which way we can interpret this procedure as an extension to systems of polynomial equations of the Gaussian elimination algorithm. In Gauss algorithm, a succession of invertible linear transformations reduces a general system of linear equations into one which is triangular. The first equation involves all the variables, the second equation does not involve 𝑥 1 , the third equation does not involved 𝑥 1 , 𝑥 2 , . . . . In the nonlinear case a similar process occurs somehow but requires, to be described correctly, to define the notion of ideal.

An Remark A.1. The Gröbner bases are exact when computed over the field Q of rational numbers.

It is a natural question whether or not, one could work with Groebner bases with coefficients in the field of real or complex numbers, or to be more exact, using floating numbers. In practice, this is a difficult topic since there are convergence/accuracy issues. Anyway, this subject is a research area called Groebner bases with coefficients in an inexact field. We redirect the interested reader to [49].

Appendix B. Upper bounds estimates of the distance to cubic elasticity Baerheim [11] has observed that the dilatation tensor d 0 = tr 12 E 0 , and the Voigt tensor v 0 = tr 13 E 0 , both second-order covariants of E 0 , generically carry information related to a "natural" basis of an elasticity tensor. Using accordingly an eigenbasis of

t 0 = A 0 : 1 - 1 4
(1 : A 0 : 1) 1, such as t ′ 0 = 2 3

( d ′ 0 -v ′ 0 ) ,
built from the asymmetric part A 0 = E 0 -(E 0 ) 𝑠 (in the sense of Backus [10]) of the raw/measured elasticity tensor, Stahn and coworkers astutely define an upper bound estimate 𝑀 (E 0 , [𝐺]) of the distance to a 3D elasticity symmetry class [𝐺], in a few steps [59]:

(1) by computing an eigenbasis of t 0 and a rotation 𝑄 0 that brings it into its diagonal form (i.e., such that 𝑄 0 ⋆ t 0 = 𝑄 0 t 0 𝑄 𝑇 0 and 𝑄 0 ⋆ t ′ 0 are diagonal), (2) by computing 𝑄 0 ⋆ E 0 , (3) finally, by setting Remark B.1. The definition (40) does not depend on a particular choice of the rotation 𝑄 0 , among the 24 possibilities bringing t 0 into its diagonal form.

It is here important to point out that difficulties arise for cubic symmetry as the second-order covariants c(E) of a cubic tensor E (such as 𝑄 ⋆ c(E) = c(𝑄 ⋆ E) ∀𝑄 ∈ SO(3)) are all isotropic, i.e. spherical. Indeed, they inherit the cubic symmetry group of E, and a cubic second-order order tensor is necessarily isotropic. This point is discussed in [11,52,2,59]. For a material with a close to be cubic microstructure, it is expected that, even if the measured tensor t 0 is orthotropic, the related eigenbasis information may be not relevant and may be of the order of magnitude of the measurement noise. This is why, starting from the harmonic decomposition of a given triclinic elasticity tensor E 0 = (𝜆 0 , 𝜇 0 , d ′ 0 , v ′ 0 , H 0 ), we propose another upper bound estimate of the distance of E 0 to elasticity cubic symmetry, as [(

d 2 2 × d 2 ) • ( d 2 2 × d 2 )] ′ ‖d 2 2 × d 2 ‖ 2
, with d 2 = H 0 . . . H 0 , (•) ′ denoting the harmonic fourth-order part. Both E upper and ∆(E 0 , [O]) are thus built from a cubic fourth-order covariant (C 0 ) defined from the fourth-order harmonic part H 0 of the raw tensor E 0 (C 0 is naively assumed to better carry the cubic basis than d 0 , v 0 or t 0 ). Component formulas for the generalized cross product d ) is cubic but not harmonic [51]. Using (18), an invariant formula for the upper bound estimate ( 41)-( 42) is finally (43) ∆(E 0 ,

[O]) = √ 2 21 ‖d ′ 0 + 2v ′ 0 ‖ 2 + 4 3 ‖d ′ 0 -v ′ 0 ‖ 2 + ‖H 0 ‖ 2 -(C 0 :: H 0 ) 2 .
The upper bound estimates ( 40) and ( 43) as well as the relative estimates

𝑀 (E 0 , [O]) ‖E 0 ‖ and ∆(E 0 , [O]) ‖E 0 ‖
are compared in Table 1 with the exact distance 𝑑(E 0 , cubic symmetry) and with the relative distance 𝑑(E 0 , cubic symmetry)/‖E 0 ‖ computed in section 6 for the raw elasticity tensor (26).

The upper bounds estimates are found not very accurate, the proposed upper bound estimate ∆(E 0 , [O]) (built from H 0 ) being only slightly better than the estimate 𝑀 (E 0 , [O]) (built from d ′ 0v ′ 0 )). This shows the difficulty to define an accurate estimate of the distance to cubic elasticity from second-order covariants of the raw elasticity tensor, and underlines the interest of being able to compute the exact distance 𝑑(E 0 , cubic symmetry). A.0.1 Symmetry classes of type I

𝑑(E

[𝑁 ] Z 2 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸1111 𝐸1122 𝐸1133 0 0 𝐸1112 𝐸1122 𝐸2222 𝐸2233 0 0 𝐸2212 𝐸1133 𝐸2233 𝐸3333 0 0 𝐸3312 0 0 0 𝐸2323 𝐸2331 0 0 0 0 𝐸2331 𝐸1313 0 𝐸1112 𝐸2212 𝐸3312 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 𝑃123 𝑃113 0 0 0 0 𝑃223 𝑃213 0 𝑃311 𝑃322 𝑃333 0 0 𝑃312 è , Ö 𝑆11 𝑆12 0 𝑆12 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] Z 3 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸2233 𝐸3112 -𝐸2312 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸2233 -𝐸3112 𝐸2312 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 𝐸3112 -𝐸3112 0 𝐸2323 0 𝐸2312 -𝐸2312 𝐸2312 0 0 𝐸2323 𝐸3112 0 0 0 𝐸2312 𝐸3112 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö -𝑃212 𝑃212 0 -𝑃213 𝑃223 -𝑃222 -𝑃222 𝑃222 0 𝑃223 𝑃213 𝑃212 𝑃322 𝑃322 𝑃333 0 0 0 è , Ö 𝑆22 0 0 0 𝑆22 0 0 0 𝑆33 èè [𝑁 ] Z 4 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 𝐸1122 𝐸2233 0 0 -𝐸2212 𝐸1122 𝐸2222 𝐸2233 0 0 𝐸2212 𝐸2233 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸2323 0 -𝐸2212 𝐸2212 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 -𝑃213 𝑃223 0 0 0 0 𝑃223 𝑃213 0 𝑃322 𝑃322 𝑃333 0 0 0 è , Ö 𝑆22 0 0 0 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] D 2 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸1111 𝐸1122 𝐸1133 0 0 0 𝐸1122 𝐸2222 𝐸2233 0 0 0 𝐸1133 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 𝑃123 0 0 0 0 0 0 𝑃213 0 0 0 0 0 0 𝑃312 è , Ö 𝑆11 0 0 0 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] D 3 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸2233 𝐸3112 0 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸2233 -𝐸3112 0 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 𝐸3112 -𝐸3112 0 𝐸2323 0 0 0 0 0 0 𝐸2323 𝐸3112 0 0 0 0 𝐸3112 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö -𝑃212 𝑃212 0 -𝑃213 0 0 0 0 0 0 𝑃213 𝑃212 0 0 0 0 0 0 è , Ö 𝑆22 0 0 0 𝑆22 0 0 0 𝑆33 èè [𝑁 ] D 4 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 𝐸1122 𝐸2233 0 0 0 𝐸1122 𝐸2222 𝐸2233 0 0 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 -𝑃213 0 0 0 0 0 0 𝑃213 0 0 0 0 0 0 0 è , Ö 𝑆22 0 0 0 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] SO(2) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸1133 0 0 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸1133 0 0 0 𝐸1133 𝐸1133 𝐸3333 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 -𝑃213 𝑃113 0 0 0 0 𝑃113 𝑃213 0 𝑃311 𝑃311 𝑃333 0 0 0 è , Ö 𝑆11 0 0 0 𝑆11 0 0 0 𝑆33 èè [𝑁 ] O(2) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸1133 0 0 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸1133 0 0 0 𝐸1133 𝐸1133 𝐸3333 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0

A.0.2 Symmetry classes of type II

The normal form of a 𝒫iez tensor belonging to a symmetry class of type II can be deduced from the previous subsection using the following lemma.

Lemma A.0.1. Let 𝐻 be a subgroup of type I. We have Fix 𝒫iez (𝐻 ⊕ Z 𝑐 2 ) = Fix Ela⊕S (𝐻).

Proof. The result follows using the harmonic decomposition of 𝒫iez (deduced from the harmonic decomposition of Piez ( [START_REF] Olive | Symmetry classes in piezoelectricity from second-order symmetries[END_REF]), Ela and S ( [START_REF] Olive | Harmonic factorization and reconstruction of the elasticity tensor[END_REF]))

𝒫iez = H 4 ⊕ H 3 ⊕ 3H 2 ⊕ H -2 ⊕ 2H 1 ⊕ 3H 0 ,
and the following two assertions:

1. Let 𝐺 be a group and (𝑉, 𝜌) be a linear representation such that 𝑉 = 𝑉 1 ⊕ 𝑉 2 stable. Then for all 𝐻 subgroup of 𝐺 Fix 𝑉 (𝐻) = Fix 𝑉 1 (𝐻) ⊕ Fix 𝑉 2 (𝐻). We can deduce from the previous lemma that

A.0.3 Symmetry classes of type III

[𝑁 ] Z -

2 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸1111 𝐸1122 𝐸1133 0 0 𝐸1112 𝐸1122 𝐸2222 𝐸2233 0 0 𝐸2212 𝐸1133 𝐸2233 𝐸3333 0 0 𝐸3312 0 0 0 𝐸2323 𝐸2331 0 0 0 0 𝐸2331 𝐸1313 0 𝐸1112 𝐸2212 𝐸3312 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 𝑃111 𝑃122 𝑃133 0 0 𝑃112 𝑃211 𝑃222 𝑃233 0 0 𝑃212 0 0 0 𝑃323 𝑃313 0 è , Ö 𝑆11 𝑆12 0 𝑆12 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] Z - 4 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 𝐸1122 𝐸2233 0 0 -𝐸2212 𝐸1122 𝐸2222 𝐸2233 0 0 𝐸2212 𝐸2233 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸2323 0 -𝐸2212 𝐸2212 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 𝑃213 -𝑃223 0 0 0 0 𝑃223 𝑃213 0 -𝑃322 𝑃322 0 0 0 𝑃312 è , Ö 𝑆22 0 0 𝑆22 0 0 𝑆33 èè [𝑁 ] Z - 6 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸2233 0 0 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸2233 0 0 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö -𝑃212 𝑃212 0 0 0 -𝑃222 -𝑃222 𝑃222 0 0 0 𝑃212 0 0 0 0 0 0 è , Ö 𝑆22 0 0 𝑆22 0 0 𝑆33 èè [𝑁 ] D 𝑧 2 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸1111 𝐸1122 𝐸1133 0 0 0 𝐸1122 𝐸2222 𝐸2233 0 0 0 𝐸1133 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 0 𝑃113 0 0 0 0 𝑃223 0 0 𝑃311 𝑃322 𝑃333 0 0 0 è , Ö 𝑆11 0 0 0 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] D 𝑧 3 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸2233 𝐸3112 0 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸2233 -𝐸3112 0 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 𝐸3112 -𝐸3112 0 𝐸2323 0 0 0 0 0 0 𝐸2323 𝐸3112 0 0 0 0 𝐸3112 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 0 𝑃223 -𝑃222 -𝑃222 𝑃223 0 𝑃223 0 0 𝑃322 𝑃322 𝑃333 0 0 0 è , Ö 𝑆22 0 0 𝑆22 0 0 𝑆33 èè [𝑁 ] D 𝑧 4 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 𝐸1122 𝐸2233 0 0 0 𝐸1122 𝐸2222 𝐸2233 0 0 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 0 𝑃223 0 0 0 0 𝑃223 0 0 𝑃322 𝑃322 𝑃333 0 0 0 è , Ö 𝑆22 0 0 0 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] D 𝑑 2 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸1111 𝐸1122 𝐸1133 0 0 0 𝐸1122 𝐸2222 𝐸2233 0 0 0 𝐸1133 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 𝑃111 𝑃122 𝑃133 0 0 0 0 0 0 0 0 𝑃212 0 0 0 0 𝑃313 0 è , Ö 𝑆11 0 0 0 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] D 𝑑 4 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 𝐸1122 𝐸2233 0 0 0 𝐸1122 𝐸2222 𝐸2233 0 0 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 𝑃213 0 0 0 0 0 0 𝑃213 0 0 0 0 0 0 𝑃312 è , Ö 𝑆22 0 0 0 𝑆22 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] D 𝑑 6 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸2233 0 0 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸2233 0 0 0 𝐸2233 𝐸2233 𝐸3333 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸2323 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö -𝑃212 𝑃212 0 0 0 0 0 0 0 0 0 𝑃212 0 0 0 0 0 0 è , Ö 𝑆22 0 0 0 𝑆22 0 0 0 𝑆33 èè [𝑁 ] O -= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 𝐸1133 𝐸1133 0 0 0 𝐸1133 𝐸2222 𝐸1133 0 0 0 𝐸1133 𝐸1133 𝐸2222 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1313 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0 0 0 𝑃312 0 0 0 0 0 0 𝑃312 0 0 0 0 0 0 𝑃312 è , Ö 𝑆33 0 0 0 𝑆33 0 0 0 𝑆33 è ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ [𝑁 ] O(2) -= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐸2222 -2𝐸1212 + 𝐸2222 𝐸1133 0 0 0 -2𝐸1212 + 𝐸2222 𝐸2222 𝐸1133 0 0 0 𝐸1133 𝐸1133 𝐸3333 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1313 0 0 0 0 0 0 𝐸1212 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Ö 0

  -Sur-Yvette Desmorat, Boris Kolev et Fabien Priziac. Je m'estime profondément chanceuse à avoir eu l'opportunité de travailler sous votre direction pendant ces quatre ans. C'est avant tout grâce à votre encadrement exemplaire que les quatre ans de thèse qui s'achèvent ont été pour moi si heureuses et stimulantes. Julien, je te remercie pour ta bienveillance, ta gentillesse et ton approche stimulante des mathématiques qui m'ont permis de découvrir la recherche avec joie et enthousiasme. Je te remercie pour ton bon humour surtout dans les moments difficiles. Rodrigue, je te remercie pour la confiance que tu m'as accordée dès le début de ma thèse, tes conseils toujours avisés tant sur le plan humain que scientifique. Je te remercie surtout pour tes encouragements qui m'ont permis de rester motivée tout au long du chemin. Boris, je suis heureuse et fière d'avoir travaillé ces quatre ans sous ta direction. Je te remercie pour tes précieux conseils, remarques, discussions, relectures et toutes les bonnes idées que tu as eues en conduisant. Enfin, Fabien, malgré la distance, tu as joué un rôle très grand dans la réussite de cette thèse. Je te remercie infiniment pour ta disponibilité à chaque fois que j'en ai eu besoin, de ta patience, de tes relectures attentives et tes conseils très précieux. J'espère que ces quelques mots ont bien exprimé mon estime et ma plus sincère reconnaissance à vous quatre. Je remercie ensuite Irina Kogan et Nicolas Dutertre pour avoir accepté d'être rapporteurs de cette thèse. Merci infiniment pour votre relecture attentive et vos retours bienveillants. Je suis également reconnaissante envers Nicolas Auffray et Jacky Cresson qui ont accepté d'être mes examinateurs. C'est un honneur pour moi de soutenir devant un tel jury.

  les filles : Roxane, Myriam, Takwa et Mihaja. Merci pour votre amitié et votre soutien tout au long de cette thèse, merci pour les longues soirées de gossip et les fous rires sponsorisé par les blagues pourris de Roro. Ensuite, je remercie mon cobureau Ahmed: merci pour ton amitié, ta gentillesse et ton bon coeur, merci pour l'ambiance musicale de travail sponsorisé par Fairouz et d'avoir ruiné ma vision à cause du manque de la lumière dans le bureau. Je remercie également Matthew pour tous les ateliers chocolat et ses anecdotes de voyage, Paul William pour tous les rires et les bons moments surtout en Belgique. Merci à Julien et Maxence pour toutes les aventures aux conférences du GDR, merci à Cedric, Corentin, Marcello, Maxime, Louis, Fatima et tous les doctorants du LMPS avec lesquelles j'ai passé des moments agréables et chaleureuses. Je passe maintenant au couloir 15-16 5ème étage, le couloir des doctorants de l'IMJ. Je commence par les meilleurs collègues de bureau : Anna, qui a été à mes côtés dans tous les sens à tous les moments, Pietro, qui ne manque aucune occasion pour me faire sentir précieuse et méritante et Haowen, qui nous n'a jamais déçu avec ces suggestions de restaurants chinois. Ensuite, je remercie tous les anciens doctorants avec lesquelles j'ai commencé ma thèse : Merci à Mahya qui a été la première à m'accueillir avec son grand sourire et qui a été une amie exceptionnelle avec laquelle j'ai partagé pleins de bons moments, Merci à Grace pour toutes les pauses et les discussions de nutritions qu'on a eu, merci aussi d'avoir partagé avec moi les malheurs de notre cher Liban, Merci à petit Thomas pour toutes les rigolades et les nombreuses pauses café, Merci à Thibaut pour toutes nos discussions de shawarma, Merci à Adrien R. pour tous nos rires, Merci à Christina, à grand Thomas, à Christophe, à Mathieu. . . Je remercie également tous les Brésiliens qui ont envahi le couloir ces dernières années et qui ont ajouté une super ambiance au labo : Thiago, João, Odylo et surtout Carla qui n'a rien avoir avec les math. Sans oublier mon brésilien préféré : Nelson, tu étais un véritable ami qui me soutenait constamment et m'encourageait dans toutes les situations, je te remercie pour ton amitié, pour toutes nos discussions de carbohydrates et surtout pour tous les moments agréables et les fous rires qu'on a partagés ensemble. Je remercie aussi mon uruguayen préféré : Joaco, je ne te connais pas depuis très longtemps, mais le temps qu'on a passé ensemble jusqu'à maintenant me suffit pour dire que j'ai gagné un ami à vie. Finalement, je remercie Chiara pour sa bonté, Raphaël en tant que membre du BDD, Jacques pour sa passion en math, Germain pour son humour, Eva pour sa gentillesse, Lucas et Mattias pour leur Cactus, Camille pour son dynamisme, Juan Felipe pour son ambiance musical, Maxime, Arnaud, Tristan, Mingchen, Cristobal, Sergi et tous ceux que j'ai omis de citer. Je tiens à remercier aussi tous les étudiants de Master qui ont passé par ce couloir notamment Adrien et Bassam. Merci à tous pour cette chaleureuse ambiance que vous avez su créer. Une pensée pour mes amis libanais à Paris qui m'ont accompagné ces années. Je remercie infiniment Christophe qui était le premier à m'encourager à venir continuer mes études en France, Merci pour ton amitié, ta générosité et ton soutien à chaque fois que j'en ai eu besoin. Je te souhaite que du bonheur pour toi et Elsa. Je remercie aussi Fabien malgré ses photos sur la plage en plein milieu de janvier, Christelle et Christian (a.k.a Fatboy). Je remercie également mes amis que j'ai rencontrés au sein de la paroisse Notre Dame du Liban : Natho (a.k.a. Samira), Christine, Danielle, Imad, Patrick, Tony (a.k.a. ma bi heb yza2ef), Aqui (a.k.a El jara), Rebecca, Myriam.

Résumé

  Cette étude tourne autour des représentations linéaires des groupes de Lie compacts et de l'étude des classes d'isotropie de ces représentations. La motivation initiale de ce travail réside dans la résolution de certains problèmes en mécanique des matériaux où des représentations de groupes compacts, tels que les sous-groupes de O(3), sont impliqués. Les classes d'isotropie des représentations de groupes sont connues pour être des ensembles semialgébriques caractérisés par des inégalités d'invariants polynomiaux. Dans cette thèse, nous prouvons que pour toute représentation d'un groupe de Lie compact, les classes d'isotropie fermées sont des ensembles algébriques. Ceci est motivé par de nombreux exemples en mécanique. Par exemple, les classes d'isotropie de la représentation de SO(3) sur l'espace des tenseurs harmoniques d'ordre 2 et 4 en mécanique sont données par des équations polynomiales, ce qui en fait des ensembles algébriques. De plus, nous prouvons un résultat en théorie des invariants, concernant la rationalité des invariants pour les représentations des groupes de Lie compacts. Plus précisément, pour toute représentation d'un groupe de Lie compact 𝐺 sur un espace vectoriel réel 𝑉 et pour tout groupe d'isotropie 𝐻, tout invariant polynomial de la représentation du normalisateur 𝑁 (𝐻) sur l'ensemble des points fixes 𝑉 𝐻 est bien la restriction sur 𝑉 𝐻 d'un invariant rationnel de la représentation initiale de 𝐺 sur 𝑉 . Cette dernière affirmation était déjà connue sous certaines conditions restrictives, comme dans le cas de corps algébriquement clos et d'isotropies génériques. Dans ce travail, les résultats sont prouvés sur R et sans autres conditions. Dans un deuxième temps, une méthode pour déterminer les classes d'isotropie pour les représentations réductibles est proposée. Cette méthode est basée sur une opération binaire entre les classes de conjugaison des sous-groupes fermés d'un groupe compact, qui a été introduite pour obtenir les classes d'isotropie d'une somme directe de représentations, si les classes d'isotropie sont connues pour les facteurs irréductibles. Dans le cas spécifique du groupe orthogonal tridimensionnel O(3), les clips entre certains types de sous-groupes de O(3) ont déjà été calculés. Cependant, jusqu'à présent, il manquait les clips entre les sous-groupes de type II et de type III. Ceux-ci apparaissent dans les lois constitutives couplées en 3D. Dans cette thèse, nous complétons les tables de clips en calculant les clips manquants. Comme application, nous obtenons 25 classes d'isotropie pour la représentation standard O(3) sur la loi piézoélectrique 3D complète, qui implique les trois tenseurs constitutifs d'élasticité, de piézoélectricité et de permittivité. Un autre problème rencontré à de nombreuses reprises en mécanique est résolu dans ce travail en utilisant des méthodes algébriques : étant donné un matériau expérimental sans isotropie spécifique, il est intéressant de déterminer l'isotropie la plus proche d'un tenseur constitutif pour ce matériau. Pour résoudre un tel problème, nous avons utilisé des méthodes d'optimisation Introduction This PhD thesis is part of the project 80PRIME GAMM (Géométrie algébrique complexe/réelle et mécanique des matériaux) co-funded by two institutes of French CNRS, (𝑖) the mathematics institute INSMI (Institut National des Sciences Mathématiques et de leurs Interactions ) and (𝑖𝑖) the engineering institute INSIS (Institut des Sciences de l'Ingénierie et des Systèmes ).

For

  this representation, there are exactly three isotropy strata of representative groups D 2 (three distinct eigenvalues), O(2) (two distinct eigenvalues) and SO(3) (only one eigenvalue). The fixed point set for the second isotropy subgroup O(2) is provided by the linear subspace 𝑉 O(2) = { 𝐴 ∈ 𝑉 ; 𝑔𝐴𝑔 -1 = 𝐴, ∀𝑔 ∈ O(2) } = {diag(-𝜆, -𝜆, 2𝜆); 𝜆 ∈ R} .

3 )

 3 using a theorem (3.5.2) that helped us reduce the complexity of the calculations. Using these results, we deduce the isotropy classes of any reducible O(3)-tensor representation and we illustrate this in the case of Piezoelectricity coupled law in chapter 4. In chapter 4, we use the clips calculations of chapter 3 to deduce the isotropy classes in the case of the coupled Piezoelectricity law which involves O(3)-representations on three tensor spaces: elasticity, piezoelectricity and permittivity. We start by introducing the Piezoelectricity coupled law and then we give a detailed description of the O(3)-representation on elasticity and piezoelectricity tensor spaces. Using the isotropy classes of each of the three representations involved and the clips operation calculated in chapter 3, we find 25 isotropy classes (theorem 4.4.1) for the Piezoelectricity coupled law.

7 .

 7 When the group 𝐺 is finite, the invariant scalar product defined in the previous proof becomes, if 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 ∈ 𝑉 , < 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 >:= 1 |𝐺| ∑ 𝑔∈𝐺 (𝑔 • 𝑣 𝑣 𝑣, 𝑔 • 𝑤 𝑤 𝑤) where |𝐺| denotes the number of elements of 𝐺. Example 1.2.8. Consider the representation 𝜌 of the group 𝐺 = SO(𝑛) on the space of real symmetric matrices 𝑉 defined by 𝜌(𝑔)𝐴 = 𝑔𝐴𝑔 -1 , for 𝑔 ∈ SO(𝑛) and 𝐴 ∈ 𝑉.

Example 3 . 1 .

 31 If 𝐺 = U(1), 𝒜 is the algebra of trigonometric polynomials R[cos(𝜃), sin(𝜃)]. Its complexification is the algebra C[cos(𝜃), sin(𝜃)]. Given a complex polynomial 𝑃 (cos 𝜃, sin 𝜃), we can associate the Laurent Polynomial 𝑃

)

  If 𝜄 denotes the Cartan involution 𝑀 → (𝑀 * ) -1 of GL(𝑛, C), then for any 𝑀 in GL(𝑚, C) with polar decomposition 𝑔ℎ, ℎ 2 = 𝜄(𝑀 ) -1 𝑀 . (3) U(𝑛) = {𝑀 ∈ GL(𝑛, C), 𝜄(𝑀 ) = 𝑀 } and P(𝑛) = {𝑀 ∈ GL(𝑛, C), 𝜄(𝑀 ) = 𝑀 -1 }. (4) The map 𝑍 → 𝑒 𝑖𝑍 from u(𝑛) to P(𝑛) is a diffeomorphism, where u(𝑛) = Lie(U(𝑛)) is the Lie algebra of skew-hermitian matrices.

Corollary 3 . 17 .

 317 It follows from lemma A.3 that 𝑔 normalizes 𝐻 C , which implies that (𝐺 C ) 𝑤 = 𝐻 C . □ For any isotropy 𝐻, the open fixed locus ∘ 𝑉 𝐻 contains a nonempty (real) Zariski open set. Proof. Thanks to corollary 3.8 we have (𝑉 𝐻 ) C = (𝑉 C ) 𝐻 = (𝑉 C ) 𝐻 C . Then the result follows from proposition 3.16 and corollary A.7.

Case 1 :

 1 the representation 𝜌 : g → o(𝑉 ) is trivial. Then, the representation itself is trivial on the neutral component of 𝐺, which means that 𝐺 factors to a representation of the finite group 𝐺/𝐺 𝑒 . Let Γ be the image of the representation, it is a finite group acting faithfully on 𝑉 . Principal points are exactly the complement of the finite union of the fixed loci 𝑉 𝛾 for 𝛾 in Γ ∖ {𝑒}. It is open and dense, we denote it by 𝑈 . Let us now partition Γ ∖ {𝑒} as Γ 1 ∪ Γ 2 where Γ 1 = {𝛾 ∈ Γ, codim 𝑉 𝛾 = 1} , and Γ 2 = {𝛾 ∈ Γ, codim 𝑉 𝛾 ≥ 2} . Let us write Γ 1 = {𝛾 1 , . . . , 𝛾 𝑝 }, Γ 2 = {𝛾 𝑝+1 , . . . , 𝛾 𝑝+𝑞 }, and let us define inductively open subsets (Ω 𝑖 ) 1≤𝑖≤𝑝+𝑞 of 𝑉 as follows: Ω 0 = Ω and Ω

  3) and SO(3)-representations on space of tensors Any finite-dimensional representation of a compact Lie group has a finite number of isotropy classes. An interesting problem would be to find the isotropy classes of a group representation. In the specific case of the three dimensional group O(3), there exist effective methods to obtain the isotropy classes of irreducible representations, and also of some reducible ones. We extend here such results to all reducible O(3)-representations using clips operation. The calculation done in this chapter is contained in an article accepted for publication in Mathematics and Mechanics of Complex Systems([5]).

. 1 )

 1 𝜌 𝑛 is called the standard representation. Thanks to the Euclidean scalar product on R 3 , we do not have to distinguish between covariant and contravariant tensors. As for the full orthogonal group O(3) = SO(3) ⊕ {±I}, another representation arises according to the possibility of -I acting trivially or as minus the identity. When -I acts trivially, the representation of O(3) on T 𝑛 (R 3 ) is reduced to the representation of SO(3) and when -I acts as minus the identity, we consider the following representation ρ𝑛 (𝑔)T = det(𝑔)𝜌 𝑛 (𝑔)T. (3.2) ρ𝑛 is called the twisted representation and such a tensor T is called a pseudo-tensor. Note that the representation of even degree 𝜌 2𝑝 of O(3) on a tensor space is reduced to the representation of SO(3) ((𝜌 2𝑛 , O(3)) is equivalent to (𝜌 2𝑛 , SO(3))). In this case, the representation of O(3) is only reduced to the twisted representation.

3 . 1 Closed

 31 O(3)-subgroupsIn the following, the notation r(𝑛 𝑛 𝑛, 𝜃), with 𝜃 ∈ [0; 2𝜋[ and 𝑛 𝑛 𝑛 = (𝑛 𝑥 , 𝑛 𝑦 , 𝑛 𝑧 ), a unit vector, denotes the Rodrigues formula to represent a rotation by angle 𝜃 around 𝑛 𝑛 𝑛, which is given by r(𝑛 𝑛 𝑛, 𝜃) = exp(𝜃𝑗(𝑛 𝑛 𝑛)) = I + 𝑗(𝑛 𝑛 𝑛) sin(𝜃) + 𝑗(𝑛 𝑛 𝑛) 2 (1cos(𝜃))

  . A subgroup of type I is conjugate to one of the following list SO(3), O(2), SO(2), D 𝑛 , Z 𝑛 , T, O, I, or 1 Type II: A closed subgroup Γ of O(3) is of type II if -I ∈ Γ, in which case -Γ -= Γ + and Γ = Γ + ∪ -Γ + . A subgroup of type II is a direct product of a subgroup of type I and the subgroup Z 𝑐 2 = {±I}, it is a conjugate to one in the following list

Figure 3 . 1 :

 31 Figure 3.1: Diagram of SO(3)-subgroups.

  .10) with b 𝑖 being the secondary axis of D 𝑛 in the (𝑥𝑦)-plane such that b 1 = 𝑒 𝑒 𝑒 1 and b 𝑖 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑛 ) b 𝑖-1

  An action of SO(3) (or O(3)) on ℋ 𝑛 (R 3 ) is induced by the natural action on R 3 : for 𝑝 ∈ ℋ 𝑛 (R 3 ) and 𝑔 ∈ O(3), (𝜌 𝑛 (𝑔)𝑝)(𝑥 𝑥 𝑥) = 𝑝(𝑔 -1 𝑥 𝑥 𝑥).

  3 and 3.4, we give the isotropy classes for SO(3)-representation 𝜌 𝑛 on the irreducible spaces H 𝑛 (R 3 ), for 𝑛 = 3, 4, 5 deduced from[53, theorem 6.6]. For the case 𝑛 = 3, an error in[53] has been corrected by Olive et al. in[START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]. In fact, in[53, theorem 6.6], [D 2 ] is said to be an isotropy class for SO(3)-representation on H 3 (R 3 )

Figure 3 . 2 :

 32 Figure 3.2: The poset of symmetry classes for H 3 .

Figure 3 . 3 :

 33 Figure 3.3: The poset of symmetry classes for H 4 .

4 .

 4 If H ∈ Σ [D 3 ] then H ∈ Σ [D 3 ] and hence d 2 ∈ Σ O(2) , tr(H × d 2 ) ̸ = 0 and (H : d 2 ) × d 2 = 0 by [91, corollary 9.6]. By [91, theorem 8.5], d 2 ∈ Σ [O(2)] gives the equation d 2 × d 2 2 = 0. As for the last equation, it is deduced from [2, proposition 4.3]. Conversely, if d 2 × d 2 2 = 0 then d 2 is either isotropic or transversely isotropic [91, lemma 8.1]. If d 2 is isotropic then 𝐻 ∈ Σ [SO(3)] ⊂ Σ [D 3 ] (see figure 3.3). If d 2 is transversely isotropic then we consider two cases:

0 and d 2 × 1

 21 c 3 = 0 then (d 2 , c 3 ) is transversely isotropic ([91, theorem 8.5]) and hence H ∈ Σ [D 4 ] or H ∈ Σ [D 3 ] or H ∈ Σ [O(2)] ([91, theorem 9.4]). But the equation 3𝐽 4 -𝐽 2 2 = 0 prevents H from being trigonal ([2, proposition 4.7]) so that in all cases we have H ∈ Σ [D 2 ] . Isotropy classes for a direct sum of irreducible representations After considering the problem of finding the isotropy classes for irreducible representations, many were interested in solving the problem for a direct sum of irreducible representations. In particular, they have established the isotropy classes for SO(3) and O(3) tensorial representations (used to model constitutive laws in mechanics), based on the fact that every finite dimensional representation of a compact Lie group can be decomposed into a direct sum of irreducible repre-

Z 3

 3 Chossat [18] omits the conjugacy class[Z 3 ]. Indeed, [Z 3 ] ∈ [T] ⊚ [O] since the intersection O ∩ 𝑔T𝑔 -1 , for 𝑔 = r ( 𝑒 𝑒 𝑒 2 -𝑒 𝑒 𝑒 3 , arccos(-13 )

  4.2). Indeed, the classes for the clips operation between a subgroup 𝐻 ⊕ Z 𝑐 2 of type II and a subgroup Γ of type III are deduced from the knowledge of the tables of the clips [Γ + ] ⊚ [𝐻] and [ Γ] ⊚ [𝐻]. By eliminating the classes that cannot be realized by any 𝑔 ∈ SO(3), we deduce the clips operation between [Γ] and [𝐻 ⊕ Z 𝑐 2 ].

  Let us consider decomposition(3.11) of the group T. Then the only non-trivial cases are obtained for 𝑔𝑒 𝑒 𝑒 𝑖 = ±𝑒 𝑒 𝑒 3 in which case we get 𝒵(𝑛), and for 𝑔s 𝑡 𝑗 = ±𝑒 𝑒 𝑒 3 we get Z 𝑑 3 .For the group O given in (3.12), the only non-trivial cases are for 𝑔a 𝑐 𝑘 = ±𝑒 𝑒 𝑒 3 and 𝑔s 𝑡 𝑗 = ±𝑒 𝑒 𝑒 3 which have been calculated in the case of T, and for 𝑔𝑒 𝑒 𝑒 𝑖 = ±𝑒 𝑒 𝑒 3 in which case we get Z 4 if 4 | 𝑛, Z - 4 if 𝑛 is even but 4 ∤ 𝑛 and Z - 2 otherwise. For the group I, we consider the decomposition (3.14) and we deduce the result following the same reasoning.Finally, for the subgroups SO(2) and O(2) given in (3.15) and (3.16) and as a direct consequence of (3.18), Lemma 3.5.6. For any integer 𝑛 ≥ 1 we have:

  r(b 1 , 𝜋), . . . , -r(b 𝑛 , 𝜋)} , b 1 = 𝑒 𝑒 𝑒 1 , b 𝑘 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑛 ) b 𝑘-1 , 𝑘 = 2, . . . , 𝑛 where b 𝑖 , 𝑖 ∈ {1, . . . , 𝑛}, are called secondary axes of the dihedral subgroup D 𝑛 (see [86, Appendix A]).

2 ]𝑑 2 ]𝑑 3 ]

 223 𝑛 = {r(b 𝑖 , 𝜋), 𝑖 = 1, . . . , 𝑛} and T = ⋃ 3 𝑖=1 Z 𝑒 𝑒 𝑒 𝑖 2 for a rotation 𝑔 such that ∃𝑖 = 1, . . . , 𝑛, 𝑔𝑒 𝑒 𝑒 1 = b 𝑖 (take for instance 𝑔 = r ( 𝑒 𝑒 𝑒 1 , for a rotation 𝑔 such that ∃𝑖 = 1, . . . , 𝑛, 𝑔𝑒 𝑒 𝑒 3 = 𝑒 𝑒 𝑒 3 (take for instance 𝑔 = r ( 𝑒 𝑒 𝑒 3 , for a rotation 𝑔 such that ∃𝑗 = 1, . . . , 4, 𝑔𝑠 𝑠 𝑠 𝑡 𝑗 = 𝑒 𝑒 𝑒 3 , [D 𝑧 𝑑 2 ] for a rotation 𝑔 such that ∃𝑖 = 1, . . . , 3, 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 3 and the two remaining 𝑒 𝑒 𝑒 𝑖 turn to two orthogonal b 𝑖 (exists for 𝑛 even). Lemma 3.5.10. Let 𝑛 ≥ 2 be an integer, 𝑑 2 = gcd(2, 𝑛), 𝑑 3 = gcd(3, 𝑛) and 𝑑 4 = gcd(4, 𝑛). We have

  𝑐 5 are three coplanar edge axes separated by an angle of 𝜋 3 ) turn to three ±b 𝑖 , [D 𝑧 𝑑 4 ] for 𝑔 = r 5.11. Let 𝑛 ≥ 2 be an integer, 𝑑 2 = gcd(2, 𝑛), 𝑑 3 = gcd(3, 𝑛) and 𝑑 5 = gcd(5, 𝑛).

1 ™ 1 =

 11 r(b 2𝑙 , 𝜋); 𝑘 2 = 0, . . . , 𝑛 -𝑒 𝑒 𝑒 1 and b 𝑙 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 𝑛 ) b 𝑙-1 , 𝑙 = 2, . . . , 𝑛. (3.23)

[Z 2 ]√ 3 )[Z 4 ]- 4 ]

 2344 for a rotation 𝑔 such that 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = b 2𝑙+1 for 𝑖 = 1, . . . 𝑛, [D 𝑑 2 ] for a rotation 𝑔 such that 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = 𝑒 𝑒 𝑒 3 and two other edge axes 𝑎 𝑎 𝑎 𝑐 𝑘 turn to two orthogonal axes b 2𝑙+1 (exists for 𝑛 even) for some 𝑙, [Z - 2 ] for a rotation 𝑔 such that 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = b 2𝑙 , [D 𝑧 2 ] for a rotation 𝑔 such that 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = b 2𝑙+1 and two other edge axes 𝑎 𝑎 𝑎 𝑐 𝑘 turn to two orthogonal axes b 2𝑙 for some 𝑙, [Z 𝑑 3 ] for a rotation 𝑔 such that 𝑔𝑠 𝑠 𝑠 𝑡 𝑗 = 𝑒 𝑒 𝑒 3 , [D 𝑑 3 ] or [D 𝑧 𝑑 3 ] for a rotation 𝑔 such that 𝑔𝑠 𝑠 𝑠 𝑡 𝑗 = 𝑒 𝑒 𝑒 3 and three other edge axes 𝑎 𝑎 𝑎 𝑐 𝑘 turn to three secondary axis of D 𝑛 (either three b 2𝑙+1 or three b 2𝑙 ) (one can take for instance 𝑔 = r ( 𝑒 𝑒 𝑒 3 , 𝜋 4 ) ∘ r (< 1, -1, 0 >, arccos ( 1 / ) to get D 𝑑 3 ), for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 3 for 𝑖 = 1, . . . , 3 (when 4 | 𝑛), [Z for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 3 for 𝑖 = 1, . . . , 3 (when 𝑛 is even and 4 ∤ 𝑛), [D 4 ] or [D 𝑧 4 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 3 with 4 | 𝑛 and two edge axes 𝑎 𝑎 𝑎 𝑐 𝑘 together with the remaining two 𝑒 𝑒 𝑒 𝑖 turn to four secondary axis of D 𝑛 (either four b 2𝑙+1 or four b 2𝑙 ), [D 𝑑 4 ] for the identity rotation for instance.

[Z - 2 ][Z 2 ]

 22 for a rotation 𝑔 such that 𝑔𝑤 𝑤 𝑤 𝑘 = b 2𝑙 for 𝑘 = 1, . . . , 15, for a rotation 𝑔 such that 𝑔𝑤 𝑤 𝑤 𝑘 = b 2𝑙+1 for 𝑘 = 1, . . . , 15, [D 𝑑 2 ] for a rotation 𝑔 such that 𝑔𝑤 𝑤 𝑤 𝑘 = 𝑒 𝑒 𝑒 3 and two other axes 𝑤 𝑤 𝑤 𝑘 turn to two orthogonal axes b 2𝑙+1 , [D 𝑧 𝑑 2 ] for a rotation 𝑔 such that 𝑔𝑤 𝑤 𝑤 𝑘 = 𝑒 𝑒 𝑒 3 and two other axes 𝑤 𝑤 𝑤 𝑘 turn to two orthogonal axes b 2𝑙 , [Z 𝑑 3 ] for a rotation 𝑔 such that 𝑔𝑣 𝑣 𝑣 𝑗 = 𝑒 𝑒 𝑒 3 for 𝑗 = 1, . . . , 10, [Z 𝑑 5 ] for a rotation 𝑔 such that 𝑔𝑢 𝑢 𝑢 𝑖 = 𝑒 𝑒 𝑒 3 for 𝑖 = 1, . . . , 6.

[Z - 2 ]

 2 for a rotation 𝑔 such that 𝑔b = b 2𝑘 for some 𝑘 and some b ∈ (𝑥𝑦)-plane and that does not turn 𝑒 𝑒 𝑒 3 to the other axes of D 2𝑛 (take for instance 𝑔 = r ( b 2𝑘 , 𝜋 3 ) ∘ r (𝑒 𝑒 𝑒 3 , 𝛼) where 𝛼 is the angle between b and b 2𝑘 ), [D 𝑑 2 ] for a rotation 𝑔 = r ( 𝑒 𝑒 𝑒 2 , 𝜋 2 ) for instance (for 𝑛 even there exists 𝑘 such that b 2𝑘+1 = 𝑒 𝑒 𝑒 2 ), [D 𝑧 2 ] for a rotation 𝑔 = r ( 𝑣 𝑣 𝑣, 𝜋 2 ) where 𝑣 𝑣 𝑣 is a secondary axis of D 2𝑛 orthogonal to 𝑒 𝑒 𝑒 3 and b 2𝑘 for instance, [D 𝑑 2𝑛 ] for the identity rotation for example. 3.5.4 Clips with O - First, we introduce a useful decomposition of the subgroup O -(see section 3.1), constructed from the couple (T, O) of index 2

. 25 )

 25 Lemma 3.5.19. Let 𝑚 ≥ 2 be an integer and 𝑑 ′ 𝑘 = gcd(𝑘, 𝑚) (𝑘 = 2, 3). Then we have

4 ]

 4 for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑎 𝑎 𝑎 𝑐 𝑘 ,[D 𝑧3 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑠 𝑠 𝑠 𝑡 𝑗 and three axes b of the (𝑥𝑦)-plane turn to 𝑎 𝑎 𝑎 𝑐 𝑘 (take for instance 𝑔 = r Ä < -1, 1, 0 >, arccos Ä for the identity rotation for instance. 3.5.5 Clips with O(2) - We construct O(2) -as follows O(2) -= SO(2) ∪ -(𝛾SO(2)) where 𝛾 = r(𝑒 𝑒 𝑒 1 , 𝜋) = {r(𝑒 𝑒 𝑒 3 , 𝜃), 𝜃 ∈ [0, 2𝜋], -r(b, 𝜋)} .

An explicit set of 25

 25 isotropy classes is deduced for the standard O(3)-representation on the coupled Piezoelectric law in three dimensions, where both elasticity tensor, piezoelectricity tensor and permittivity tensor occur. The result obtained in this chapter is contained in an article accepted for publication in Mathematics and Mechanics of Complex Systems ([5]).

  The eight elasticity isotropy classes for the SO(3)-representation on Ela are depicted in Figure 4.1 with the mechanical nomenclature and arrows corresponding to the partial order ⪯ (an arrow from [𝐻 1 ] to [𝐻 2 ] means that [𝐻 1 ] ⪯ [𝐻 2 ], see subsection 1.2.2).

Figure 4 . 1 :

 41 Figure 4.1: The poset of symmetry classes for Ela [53, 37].

3 . 2 )

 32 and the harmonic decomposition defined in subsection 4.3.3. By these means, the 16 symmetry classes of Piez are directly obtained in the following theorem (the notations and definitions of O(3)-subgroups are fully given in section 3.1): Theorem 4.3.1 (Olive et al. 2021 [89]). A conjugacy class of a closed subgroup of O(3) is a symmetry class of a tensor P ∈ Piez if and only if it belongs to the 16 elements set of symmetry classes

Theorem 4 . 3 . 3 .

 433 Let P = g + h ∈ Piez be a piezoelectricity tensor, with h ∈ H 3 (R 3 ) its leading harmonic part, letd 2 = h : h (i.e., (d 2 ) 𝑖𝑗 = ℎ 𝑖𝑘𝑙 ℎ 𝑘𝑙𝑗 ) ,and d ′ 2 = d 2 -1 3 tr(d 2 ) q be second-order covariants of P.Then P ∈ Σ [O -] (is at least cubic) ifand only if g = 0 and d ′ 2 = 0,and P ∈ Σ [O -] (is cubic) if and only if furthermore h ̸ = 0.The proof of this theorem is fully detailed in the appendix C of chapter 5 of this manuscript. 4.4 Isotropy classes for the coupled piezoelectricity law using clips operation between O(3)-subgroups In the following, we denote by 𝒥 (𝑉 ) the set of isotropy classes of the representation of either SO(3) or O(3) on the vector space 𝑉 . As a consequence of lemma 3.3.2, the isotropy classes 𝒥 (𝒫iez) can be deduced from isotropy classes Ela, Piez and S: 𝒥 (𝒫iez) = (𝒥 (Ela) ⊚ 𝒥 (Piez)) ⊚ 𝒥 (S). Recall from theorem 4.3.1 the isotropy classes of Piez (the notations and definitions of O(3)subgroups have been recalled in section 3.1):

[

  𝐻] := { 𝑔𝐻𝑔 -1 ; 𝑔 ∈ 𝐺 } the conjugacy class of the subgroup 𝐻 of 𝐺. To each conjugacy class [𝐻], where 𝐻 is a closed subgroup of 𝐺, corresponds the subset of V defined by Σ [𝐻] := {𝑣 𝑣 𝑣 ∈ V; [𝐺 𝑣 𝑣 𝑣 ] = [𝐻]} . If this subset is not empty, [𝐻] is called an isotropy class and Σ [𝐻] is the isotropy stratum associated to [𝐻]. It is known (see

( 2 )

 2 𝑓 * = inf {𝑓 (𝑥 𝑥 𝑥); 𝑥 𝑥 𝑥 ∈ 𝐾} , where 𝑓 ∈ R[𝑋] := R[𝑋 1 , . . . , 𝑋 𝑛 ], 𝑥 𝑥 𝑥 = (𝑥 1 , . . . , 𝑥 𝑛 ) ∈ R 𝑛 and 𝐾 is a basic closed semialgebraic set 𝐾 = {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔 1 (𝑥 𝑥 𝑥) ≥ 0, . . . , 𝑔 𝑚 (𝑥 𝑥 𝑥) ≥ 0} ,

Definition 3 . 2 .

 32 Consider the R-module M(𝑔 1 , . . . , 𝑔 𝑚 ) := { 𝜎 0 + 𝑚 ∑ 𝑖=1 𝜎 𝑖 𝑔 𝑖 ; 𝜎 0 , . . . , 𝜎 𝑚 sum of squares } ⊂ R[𝑋] (a polynomial 𝑝 ∈ R[𝑋] is a sum of squares if there exist polynomials 𝑝 1 , . . . , 𝑝 𝑁 ∈ R[𝑋] such that 𝑝 = ∑ 𝑁 𝑗=1 𝑝 2 𝑗 ). We say that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 satisfy the Archimedean hypothesis (or that M(𝑔 1 , . . . , 𝑔 𝑚 ) is an Archimedean module) if there exists a positive integer 𝑁 such that 𝑁 -𝑛 ∑ 𝑖=1 𝑋 2 𝑖 ∈ M(𝑔 1 , . . . , 𝑔 𝑚 ).

Finally

  , for 𝑖 ∈ {0, . . . , 𝑛}, denote 𝑣 𝑖 := ⌈ deg(𝑔 𝑖 ) 2⌉(notice that 𝑣 0 = 0) and let 𝑑 0 be the integer max

  The numerical problem is first reduced to the following polynomial optimization problem min a∈𝐾 ‖a 0 -a‖ 2 ,

Figure 1 .

 1 Figure 1. Cubic microstructure of CMSX-4 Ni-based single crystal superalloy [51].

Remark 7 . 4 .⎞⎠ C/m 2

 742 We take 𝑐 = 3 > 𝑓 (0) to ensure the Archimedean property, but in the present case the convergence status 𝜉 does not seem to depend on 𝑐. Dropping the condition 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0 in K also leads to an accurate computed optimum. We obtain the result min 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥) ≈ 𝑓 (𝑥 𝑥 𝑥 * ) = 1.060855 C 2 /m 4 at the first GloptiPoly relaxation 𝑑 = 𝑑 0 = 1 (with convergence status 𝜉 = +1 and for a computation time of 0.8 seconds). The components of the computed minimizer h * are (in C/m 2 ): ℎ 111 = -0.075476, ℎ 112 = -0.426450, ℎ 122 = 0.088998, ℎ 123 = -0.005937, ℎ 222 = 0.412070, ℎ 223 = -0.308913, ℎ 333 = 0.609783. By theorem 7.3, the computed cubic tensor e * ∈ Σ [O -] the closest to e 0 is simply e * = h * . In Voigt notation, [e * ] = ⎛ ⎝ -0.075476 0.088998 -0.013521 -0.005937 -0.300870 -0.426450 -0.426450 0.412070 0.0143797 -0.308913 -0.005937 0.088998 -0.300870 -0.308913 0.609783 0.014379 -0.013521 -0.005937 The distance and the relative distance to cubic piezoelectricity are finally ∆(e 0 , Σ [O -] ) ≈ ‖e 0e * ‖ = 1.214681 C/m

  𝛼 = tr(d), 𝛽 = tr(v), and

( 4 ) 6 (

 46 b) 𝑖𝑗𝑘𝑙 = (a ⊙ b) 𝑖𝑗𝑘𝑙 = 1 𝑎 𝑖𝑗 𝑏 𝑘𝑙 + 𝑏 𝑖𝑗 𝑎 𝑘𝑙 + 𝑎 𝑖𝑘 𝑏 𝑗𝑙 + 𝑏 𝑖𝑘 𝑎 𝑗𝑙 + 𝑎 𝑖𝑙 𝑏 𝑗𝑘 + 𝑏 𝑖𝑙 𝑎 𝑗𝑘 )𝑏 𝑘𝑙 + 2𝑏 𝑖𝑗 𝑎 𝑘𝑙 -𝑎 𝑖𝑘 𝑏 𝑗𝑙 -𝑎 𝑖𝑙 𝑏 𝑗𝑘 -𝑏 𝑖𝑘 𝑎 𝑗𝑙 -𝑏 𝑖𝑙 𝑎 𝑗𝑘 ).

( 4 )

 4 𝑆 := {𝑥 𝑥 𝑥 ∈ R 𝑛 ; 𝑔(𝑥 𝑥 𝑥) = 0} .

  𝑓 (𝑥 𝑥 𝑥) with 𝑔(𝑥 𝑥 𝑥) = 0, are critical points of the function (6) 𝐹 (𝑥 𝑥 𝑥, 𝜆 𝜆 𝜆) = 𝑓 (𝑥 𝑥 𝑥) + (𝜆 𝜆 𝜆, 𝑔(𝑥 𝑥 𝑥)), where (•, •) is the duality bracket on R 𝑝 and the dual variable 𝜆 𝜆 𝜆 is the Lagrange multiplier. A proof of this fact can be found in [1, Theorem 3.5.27].

( 16 ) 1 3 (Theorem 3 . 3 (

 161333 𝐸 1111 = 2𝜇 + 𝜆 + 8𝛿, 𝐸 1122 = 𝜆 -4𝛿, 𝐸 1212 = 𝜇 -4𝛿, where the invariants 𝜆, 𝜇, 𝐽 2 , 𝐽 3 and 𝛿 are evaluated on E * . The covariant characterization of the elasticity symmetry classes by polynomial equations (and inequalities) has been performed recently, in [53, theorem 10.2]. The case of the cubic symmetry is recalled as theorem 3.3 below. We denote by a ′ = a -1 3 tr(a) 1, the deviatoric part of a second-order tensor a, and by E 𝑠 , the totally symmetric part of E, with components 𝐸 𝑠 𝑖𝑗𝑘𝑙 = 𝐸 𝑖𝑗𝑘𝑙 + 𝐸 𝑖𝑘𝑗𝑙 + 𝐸 𝑖𝑙𝑗𝑘 ). Olive et al (2021)). Let E be an elasticity tensor, d = tr 12 E and v = tr 13 E, respectively, the dilatation and the Voigt second-order tensors, dtr v) and 𝜇 = 1 30 (3 tr vtr d), the Lamé constants,

Theorem 4 . 1 . 5 .

 415 Let E = (𝜆, 𝜇, d ′ , v ′ , H) ∈ Ela and F = (ℓ, 𝑚, e ′ , w ′ , K) ∈ Ela be two elasticitylike fourth-order tensors, andd 2 (H) = H . . . H, d 2 (K) = K . . . K,be the quadratic covariants of their respective harmonic fourth-order components H and K.Then, the pair (E, F) is cubic if and only ifd ′ = v ′ = e ′ = w ′ = 0,and either (𝑎) d ′ 2 (H) = 0 and H = 𝑘K ̸ = 0, or (𝑏) d ′ 2 (K) = 0 and K = 𝑘H ̸ = 0, with 𝑘 ∈ R. Distance to cubic elasticity as a quadratic optimization problem It is possible to reformulate the distance to cubic symmetry problem into a quadratic optimization problem min E ‖E 0 -E‖ 2 with E at least cubic, since the function to be minimized ‖E 0 -E‖ 2 and the constraint "E at least cubic" (equivalent to d ′ = v ′ = 0 and d ′ 2 = 0 by theorem 3.3) are expressed by quadratic polynomials in E. This 21-dimensional problem can be further reduced to a 9-dimensional optimization problem in the harmonic component H ∈ H 4 of E only.

(

  𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 )

15 ( 2

 152 tr etr w) and 𝑚 = 1 30 (3 tr wtr e), the Lamé constants of F, where e := tr 12 F and w := tr 13 F,

Lemma 7 . 3 .

 73 The first-order Euler-Lagrange equations,

5 scalar equations) Remark 7 . 4 .

 74 The distance problem thus formulated is not a quadratic optimization problem. The equation (𝑘H -K 0 ) :: H = 0 is indeed polynomial, but of degree three in the variable 𝑥 𝑥 𝑥 = (H, 𝑘).

  -54 -83 -34 -94 59 -54 176 -71 71 -40 -23 -83 -71 207 -44 130 -36 -34 71 -44 99 -15 -17 -94 -40 130 -15 179 -40 59 -23 -36 -17 -40 79

  𝑓 (H) = ‖H 0 -H‖ 2 + 𝑊 ‖K 0 -𝑘H‖ 2 ,that we have to minimize in order to solve the distance problem min (E,F) cubic

  b) the scalar equation (H -H 0 ) :: H = 0 (detailed in point (b) of section 6), (c) the scalar equation

[((((

  𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 ) 𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 ) 𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 ) 𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 )

  normal form of E * is E * O = 𝑄 ⋆ E * ,and one recovers(29).

.

  The normal forms of E ** and F ** are E ** O = 𝑄⋆E ** and F ** O = 𝑄⋆F ** . They are (simultaneously) obtained for the (same) rotation 𝑄. One then recovers the normal forms given at the end of section 8 (i.e., in Voigt notation, the matrices [E ** O ] and [F ** O ]).

  ideal 𝐼 of the algebra C[𝑥 1 , . . . , 𝑥 𝑛 ] is a subalgebra of C[𝑥 1 , . . . , 𝑥 𝑛 ] which is stable by multiplication by every polynomial in C[𝑥 1 , . . . , 𝑥 𝑛 ]. More precisely, this means that 𝑓 ∈ 𝐼 and 𝑝 ∈ C[𝑥 1 , . . . , 𝑥 𝑛 ] =⇒ 𝑝𝑓 ∈ 𝐼. Now consider the system (36) and the ideal𝐼 := {𝑝 1 𝑓 1 + • • • + 𝑝 𝑚 𝑓 𝑚 ; 𝑝 𝑘 ∈ C[𝑥 1 , . . . , 𝑥 𝑛 ]} ,generated by 𝑓 1 , . . . , 𝑓 𝑚 . It is clear that every 𝑓 ∈ 𝐼 vanishes on each solution of(36), and conversely that the solutions of (36) can be recast as the solutions of the infinite system of equations 𝑓 (𝑥 1 , . . . , 𝑥 𝑛 ) = 0, 𝑓 ∈ 𝐼. Now, let us introduce the 𝑙-th elimination ideal 𝐼 𝑙 := 𝐼 ∩ C[𝑥 𝑙+1 , . . . , 𝑥 𝑛 ], 𝑙 = 1, . . . , 𝑛 -1.

( 40 ) 24 ∑𝑖=1𝑅

 4024 𝑀 (E 0 , [𝐺]) := min 𝑄∈O ‖𝑄 ⋆ 𝑄 0 ⋆ E 0 -R 𝐺 (𝑄 ⋆ 𝑄 0 ⋆ E 0 )‖,with the property 𝑄 ⋆ 𝑄 0 ⋆ E 0 = (𝑄𝑄 0 ) ⋆ E 0 and where R 𝐺 is the Reynolds operator (an orthogonal projector [60,Chapter 2], see also[6, Appendix C]), associated with the group 𝐺 representative of the considered symmetry class. In particular, for the cubic elasticity case and the finite proper octahedral (cubic) group 𝐺 = O (of order 24), one hasR O (E) = 1 24 𝑖 ⋆ E,where the rotations 𝑅 1 = I, 𝑅 2 , . . . , 𝑅 24 are the 24 elements of the group O.

  0 , [O]) := ‖E 0 -E upper ‖, where the cubic elasticity tensor (42) E upper := 2𝜇 0 I + 𝜆 0 1 ⊗ 1 + (C 0 :: H 0 ) C 0 is computed from the cubic harmonic fourth-order tensor of unit norm C 0 := √ 15 2

èA

  basis of the fixed-point set of a subgroup 𝐻 is computed by solving a linear system defined using the generators of 𝐻 from table 3.1: if ℎ 𝑖 are the generators of 𝐻 then we have⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜌(ℎ 𝑖 )E = E ∀𝑖 𝜌(ℎ 𝑖 )P = P ∀𝑖 𝜌(ℎ 𝑖 )S = S ∀𝑖 In the following, an element of each fixed-point set is represented by a triplet of normal forms in their matrix representation. A triplet of Fix(𝐻) is denoted by [𝑁 ] 𝐻 = ([E] 𝐻 , [P] 𝐻 , [S] 𝐻 ).

2 .

 2 Fix H 𝑛 (𝐻 ⊕ Z 𝑐 2 ) = ⎧ ⎨ ⎩ Fix(𝐻) if 𝑛 is even (𝜌(-I) = I) 0 if 𝑛 is odd (𝜌(-I) = -I).Hence, we have Fix 𝒫iez (𝐻 ⊕ Z 𝑐 2 ) = Fix H 4 ⊕3H 2 ⊕3H 0 (𝐻) = Fix Ela⊕S (𝐻).

[E] 𝐻⊕Z 𝑐 2 =

 2 [E] 𝐻 , [P] 𝐻⊕Z 𝑐 2 = 0 and [S] 𝐻⊕Z 𝑐 2 = [S] 𝐻 .

  The use of the inverse is required so the first property of being an action is satisfied.

1.1 Action of a group on a set 1.1.1 Definition and properties Let 𝐺 be a group and 𝑉 a set. An action of 𝐺 on 𝑉 is a map 𝐺 × 𝑉 → 𝑉 (𝑔, 𝑣 𝑣 𝑣) ↦ → 𝑔 • 𝑣 𝑣 𝑣 satisfying the following properties 1. For all 𝑔, ℎ ∈ 𝐺 and 𝑣 𝑣 𝑣 ∈ 𝑉 , 𝑔 • (ℎ • 𝑣 𝑣 𝑣) = 𝑔ℎ • 𝑣 𝑣 𝑣; 2. For all 𝑣 𝑣 𝑣 ∈ 𝑉 , 𝑒 • 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣 where 𝑒 is the identity of 𝐺. Remark 1.1.1. This definition corresponds to a left action but a group can also acts on the right. In this manuscript, we only consider left actions. Example 1.1.2. Any group acts on itself by left multiplication. In other words, 𝐺 acts on 𝑉 = 𝐺 by 𝑔 • ℎ = 𝑔ℎ where 𝑔, ℎ ∈ 𝐺. A closely related action is given by right multiplication 𝑔 • ℎ = ℎ𝑔 -1 if 𝑔, ℎ ∈ 𝐺. Suppose that we have an action of 𝐺 on 𝑉 , we say that a subset 𝑊 of 𝑉 is 𝐺-stable if ∀𝑔 ∈ 𝐺, ∀𝑣 𝑣 𝑣 ∈ 𝑊, 𝑔 • 𝑣 𝑣 𝑣 ∈ 𝑊, (1.1) and we say that 𝑊 is 𝐺-invariant if ∀𝑔 ∈ 𝐺, ∀𝑣 𝑣 𝑣 ∈ 𝑊, 𝑔 • 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣. (1.2) Example 1.1.3. Consider the action of the rotation group 𝑆𝑂(2) on R 2 defined by 𝑔 • (𝑥, 𝑦) := (𝑥 cos(𝜃) -𝑦 sin(𝜃), 𝑥 sin(𝜃) + 𝑦 cos(𝜃)) for 𝑔 ∈ 𝑆𝑂(2) and (𝑥, 𝑦) ∈ R 2 . Any circle

  Remark 1.1.7. We consider the projection map 𝑉 is a topological set, the orbit space 𝑉 /𝐺 is equipped with the quotient topology induced from the topology on 𝑉 :𝑈 ⊂ 𝑉 /𝐺 is open if and only if 𝜋 -1 (𝑈 ) is open in 𝑉.

	The map 𝜋 is open with respect to this topology.
	1.1.3 Orbit types

𝜋 : 𝑉 → 𝑉 /𝐺.

When

If 𝐺 acts on 𝑉 and 𝑣 𝑣 𝑣 ∈ 𝑉 , the set of elements of 𝐺 fixing 𝑣 𝑣 𝑣 is a subgroup of 𝐺, called the stabilizer of 𝑣 𝑣 𝑣, and denoted by 𝐺 𝑣 𝑣 𝑣 :

𝐺 𝑣 𝑣 𝑣 := {𝑔 ∈ 𝐺; 𝑔 • 𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣} .

For the action of a topological group 𝐺 on a topological set 𝑉 , 𝐺 𝑣 𝑣 𝑣 is a closed subgroup of 𝐺 (𝐺 𝑣 𝑣 𝑣 = 𝜋 -1 𝑣 𝑣 𝑣 ({𝑣 𝑣 𝑣}) where 𝜋 𝑣 𝑣 𝑣 : 𝐺 → 𝑉 ; 𝜋 𝑣 𝑣 𝑣 (𝑔) = 𝑔 • 𝑣 𝑣 𝑣). 𝐺 𝑣 𝑣 𝑣 is called the isotropy subgroup (or symmetry group) of 𝑣 𝑣 𝑣. We can remark that isotropy groups of elements of a same orbit are conjugate: Lemma 1.1.8. For any 𝑣 𝑣 𝑣 ∈ 𝑉 and any 𝑔 ∈ 𝐺, we have

  1 ≾ 𝑣 𝑣 𝑣 2 ⇐⇒ there exists a 𝐺-equivariant map 𝜑 : 𝐺 • 𝑣 𝑣 𝑣 1 → 𝐺 • 𝑣 𝑣 𝑣 2

	(1.8)
	and we write
	𝑉

≾ 𝑣 𝑣 𝑣 = {𝑤 𝑤 𝑤 ∈ 𝑉 | 𝑣 𝑣 𝑣 ≾ 𝑤 𝑤 𝑤} . Lemma 1.1.10. [30, lemma 2.6.2] For 𝑣 𝑣 𝑣 1 , 𝑣 𝑣 𝑣 2 ∈ 𝑉 , we have 1. 𝑣 𝑣 𝑣 1 ≈ 𝑣 𝑣 𝑣 2 if and only if 𝐺 𝑣 𝑣 𝑣 1 is conjugate to 𝐺 𝑣 𝑣 𝑣 2 within 𝐺, that is, there exists 𝑔 ∈ 𝐺 such that 𝐺 𝑣 𝑣 𝑣 1 = 𝑔𝐺 𝑣 𝑣 𝑣 2 𝑔 -1 . 2. 𝑣 𝑣 𝑣 1 ≾ 𝑣 𝑣 𝑣 2 if and only if 𝐺 𝑣 𝑣 𝑣 1 is conjugate to a subgroup of 𝐺 𝑣 𝑣 𝑣 2 within 𝐺, that is, there exists 𝑔 ∈ 𝐺 such that 𝐺 𝑣 𝑣 𝑣 1 ⊂ 𝑔𝐺 𝑣 𝑣 𝑣 2 𝑔 -1 . Proof. 1. Suppose that 𝑣 𝑣 𝑣 1 ≈ 𝑣 𝑣 𝑣 2 then there exists a 𝐺-equivariant bijection 𝜑 from 𝐺 • 𝑣 𝑣 𝑣 1 to 𝐺 • 𝑣 𝑣 𝑣 2 . Then, there exists 𝑔 ∈ 𝐺 such that 𝜑(𝑣 𝑣 𝑣 1 ) = 𝑔 • 𝑣 𝑣 𝑣 2 and 𝐺 𝜑(𝑣 𝑣 𝑣 1 ) = 𝐺 𝑣 𝑣 𝑣 1 . We deduce that 𝐺 𝑣 𝑣 𝑣 1 = 𝑔𝐺 𝑣 𝑣 𝑣 2 𝑔 -1

  1 , 𝑣 𝑣 𝑣 2 ∈ 𝑉 have the same isotropy class iff ∃𝑔 ∈ 𝐺 / 𝐺 𝑣 𝑣 𝑣 2 = 𝑔𝐺 𝑣 𝑣 𝑣 1 𝑔 -1 . Given a closed subgroup 𝐻 of 𝐺, we say that the conjugacy class [𝐻] of 𝐻 in 𝐺 is an isotropy class (or an orbit type ) if there exists an element 𝑣 𝑣 𝑣 such that [𝐻] = [𝐺 𝑣 𝑣 𝑣 ]. "Having the same isotropy class" is an equivalence relation defined on 𝑉 which is the same (by lemma 1.1.10) as the equivalence relation defined in (1.6). "Having the same isotropy class" (or equivalentely the relation ≈) is coarser than the relation "being in the same orbit" (1.3) defined in the previous section. In fact, due to lemma 1.1.8, vectors of same 𝐺-orbit have the same isotropy class but the converse is not always true. Here is a counterexample. Example 1.1.11. Consider the usual action of SO(2) on R 2 . All vectors except 0 have conjugate symmetry groups but they are not in the same orbit.

For an isotropy subgroup 𝐻 of 𝐺 (i.e. 𝐻 = 𝐺 𝑣 𝑣 𝑣 for 𝑣 𝑣 𝑣 ∈ 𝑉 ), we gather the vectors having the same isotropy class in a set called the isotropy stratum with respect to 𝐻, denoted by Σ

[𝐻]

  Definition 1.2.1. A representation (𝑉, 𝐺) is said to be reducible if there exists a non trivial subspace 𝑊 of 𝑉 (that is 𝑊 ̸ = {0} and 𝑊 ̸ = 𝑉 ) that is 𝐺-stable. It is said to be irreducible if it is not reducible. 1.2.4. A decomposable representation is reducible but the converse is not always true.As a counterexample, consider the following representation of R on R 2

Example 1.2.2. The representation of the group SO(𝑛, R) on R 𝑛 given by 𝜌(𝑔)𝑣 𝑣 𝑣 = 𝑔𝑣 𝑣 𝑣, if 𝑔 ∈ SO(𝑛, R) and 𝑣 𝑣 𝑣 ∈ R 𝑛 , is irreducible. Definition 1.2.3. A representation (𝑉, 𝐺) is said to be decomposable if 𝑉 can be decomposed into a direct sum of irreducible subspaces.

Remark

  Definition 1.2.5. A group 𝐺 is said to be linearly reductive if every finite dimensional representation of 𝐺 is decomposable.

Lemma 1.2.6. Every compact group is linearly reductive.

Proof. Let 𝑈 ⊂ 𝑉 be a 𝐺-stable subspace and < •, • > be a 𝐺-invariant scalar product on 𝑉 (which exists since 𝐺 is compact; take for instance the scalar product defined, for 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 ∈ 𝑉 , by < 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 >:= ¡ 𝐺 (𝑔 • 𝑣 𝑣 𝑣, 𝑔 • 𝑤 𝑤 𝑤)𝑑𝑔 where (•, •) is a scalar product on 𝑉 and 𝑑𝑔 is the Haar measure (left and right Haar measure are the same since 𝐺 is compact) on 𝐺 (see [13, Chapter 0 section 3], [14, page 46

  Definition 1.2.12. For any 𝑣 𝑣 𝑣 in 𝑉 , a local slice of the 𝐺-action at 𝑣 is a local submanifold 𝑆 ⊂ 𝑉 passing through 𝑣 𝑣 𝑣 such that:

  2 are two points in 𝑆 and if there exists 𝑔 in 𝐺 with 𝑔 • 𝑠 1 = 𝑠 2 , then 𝑔 belongs to 𝐺 𝑣 𝑣 𝑣 . In particular, 𝐺 𝑠 ⊂ 𝐺 𝑣 𝑣 𝑣 for each point 𝑠 ∈ 𝑆. proof of this theorem is provided in appendix B of chapter 2.It follows from the connectivity of 𝑈 Ω /𝐺 that there is a unique isotropy stratum Σ [𝐻 0 ] which is minimal. This isotropy class [𝐻 0 ] is called the principal stratum. We can conclude that the principal orbit type [𝐻 0 ] is the smallest isotropy class.Example 1.2.17. Consider, for instance, the representation of the rotation group 𝐺 = SO(3)

𝐺 • 𝑆 is an open neighborhood of the orbit 𝐺 • 𝑣 𝑣 𝑣.

Slices exist in general for proper smooth actions of Lie groups on manifolds (see

[30, Theorem 2.3.1]

for the precise hypothesis on the actions), and in particular for actions of compact Lie groups. However, for linear representations of compact Lie groups, it is possible to produce slices easily (see lemma B.3 of appendix B of chapter 2). The existence of slices has the following corollary.

Corollary 1.2.13. Let 𝑣 𝑣 𝑣 in 𝑉 . Then there exists a neighborhood 𝑈 of 𝑣 𝑣 𝑣 such that for all 𝑤 𝑤 𝑤 in

𝑈 , [𝐺 𝑤 𝑤 𝑤 ] ⪯ [𝐺 𝑣 𝑣 𝑣 ].

Proof. Let 𝑆 be a slice at 𝑣 𝑣 𝑣. If 𝑤 𝑤 𝑤 ∈ 𝐺.𝑆 then 𝑤 𝑤 𝑤 is in the orbit of a point 𝑤 𝑤 𝑤 ′ in 𝑆. It follows that

𝐺 𝑤 𝑤 𝑤 ′ ⊂ 𝐺 𝑣 𝑣 𝑣 , so [𝐺 𝑤 𝑤 𝑤 ] = [𝐺 𝑤 𝑤 𝑤 ′ ] ⪯ [𝐺 𝑣 𝑣 𝑣 ].

Due to the existence of a partial order on conjugacy classes, points with locally minimal isotropy can be defined: Definition 1.2.14. A vector 𝑣 𝑣 𝑣 ∈ 𝑉 (or its orbit) is called principal if there exists a neighborhood 𝑈 of 𝑣 𝑣 𝑣 such that for all 𝑤 𝑤 𝑤 in 𝑈 , [𝐺 𝑤 𝑤 𝑤 ] = [𝐺 𝑣 𝑣 𝑣 ]. Equivalently, 𝑣 𝑣 𝑣 is principal if it has locally minimal isotropy. Remark 1.2.15. The set of principal points is open by definition. Let us denote by 𝜋 the canonical projection from 𝑉 to 𝑉 /𝐺. The principal orbit type theorem can be stated (in the case of linear representations) as follows (see [13, Theorem 3.1], [1, section VI], [30, Theorem 2.8.5]). Theorem 1.2.16 (Principal orbit type theorem). For any open subset Ω of 𝑉 such that 𝜋(Ω) is connected, the set 𝑈 Ω of principal points in Ω is open, dense in Ω and 𝑈 Ω /𝐺 is connected.

A

  20. Let 𝐻 be an isotropy subgroup. Then ∘ 𝑉 𝐻 is open and dense in 𝑉 𝐻 , more precisely it contains an open dense subset of 𝑉 𝐻 . Then 𝑔𝐻𝑔 -1 ⊂ 𝐺 𝑣 𝑣 𝑣 so that 𝑣 𝑣 𝑣 belongs to Σ [𝐻] . Now, let 𝑣 𝑣 𝑣 be in Σ [𝐻] . By definition, there exists 𝑔 in 𝐺 such that 𝑔 -1 • 𝑣 𝑣 𝑣 belongs to 𝑉 𝐻 . Thanks to proposition 1.2.20, 𝑔 -1 • 𝑣 𝑣 𝑣 is the limit of a sequence of elements in

Proof. Let 𝑥 𝑥 𝑥 be in ∘ 𝑉 𝐻 . Then thanks to Corollary 1.2.13 there exists a neighborhood 𝑈 of 𝑥 𝑥 𝑥 such

that for 𝑦 𝑦 𝑦 in 𝑈 , [𝐺 𝑦 𝑦 𝑦 ] ⪯ [𝐻]. If 𝑦 𝑦 𝑦 in 𝑈 ∩ 𝑉 𝐻 , [𝐻] ⪯ [𝐺 𝑦 𝑦 𝑦 ] ⪯ [𝐻] so [𝐺 𝑦 𝑦 𝑦 ] = [𝐻]

. This means that 𝐻 ⊂ 𝐺 𝑦 𝑦 𝑦 = 𝑔𝐻𝑔 -1 for some 𝑔 in 𝐺 so 𝐺 𝑦 𝑦 𝑦 = 𝐻 by lemma 1.2.10. This proves that

∘ 𝑉 𝐻 is open.

The second part is difficult and will be proven later (see Corollary 3.17 of chapter 2).

Remark 1.2.21. For polynomial actions, an open dense subset is also a

Zariski open 1 subset. Corollary 1.2.22. Σ [𝐻] is the closure of Σ [𝐻] . Proof. First we prove that Σ [𝐻] is closed. If (𝑣 𝑣 𝑣 𝑛 ) 𝑛∈N is a sequence in Σ [𝐻]

that converges to a vector 𝑣 𝑣 𝑣 ∈ 𝑉 then, for all 𝑛 ∈ N, there exists an element 𝑔 𝑛 in 𝐺 such that, ∀𝑛 ∈ N,

𝑔 𝑛 𝐻𝑔 -1

𝑛 ⊂ 𝐺 𝑣 𝑣 𝑣𝑛 . Extracting a subsequence, we can assume that (𝑔 𝑛 ) 𝑛∈N converges to an element 𝑔 in 𝐺. ∘ 𝑉 𝐻 .

  .10) Remark 1.2.23. Note that 𝑁 (𝐻) is the maximal subgroup of 𝐺 in which 𝐻 is a normal subgroup. Indeed, 𝑁 (𝐻) is the isotropy subgroup of 𝐻 for the action of the group 𝐺 on the set of all subgroups of 𝐺, defined by

The cardinal of the monodromy group determine the number of points in the intersection of each 𝐺-orbit of a point of ∘ 𝑉 𝐻 and 𝑉 𝐻

  Zariski open set is the complement set of zeros of a finite number of polynomials Lemma 1.2.24. For each closed subgroup 𝐻 of 𝐺, 𝑉 𝐻 is 𝑁 (𝐻)-stable. Moreover, if 𝐻 = 𝐺 𝑣 𝑣 𝑣 0 is the isotropy group of some vector 𝑣 𝑣 𝑣 0 ∈ 𝑉 , then 𝑁 (𝐻) ={ 𝑔 ∈ 𝐺; | 𝑔 • 𝑉 𝐻 ⊂ 𝑉 𝐻 } .Proof. Let 𝑣 𝑣 𝑣 in 𝑉 𝐻 . For any 𝑔 in 𝑁 (𝐻) and any ℎ in 𝐻, ℎ• (𝑔 • 𝑣 𝑣 𝑣) = 𝑔 • (𝑔 -1 ℎ𝑔) • 𝑣 𝑣 𝑣 = 𝑔 • 𝑣 𝑣 𝑣 (since 𝑔 ∈ 𝑁 (𝐻)). If moreover 𝐻 = 𝐺 𝑣 𝑣 𝑣 0 and 𝑔 ∈ 𝐺 is such that 𝑔 • 𝑉 𝐻 ⊂ 𝑉 𝐻 , then 𝑔 • 𝑣 𝑣 𝑣 0 is in 𝑉 𝐻 but 𝐺 𝑔•𝑣 𝑣 𝑣 0 = 𝑔𝐺 𝑣 𝑣 𝑣 0 𝑔 -1 and thus 𝐻 ⊂ 𝑔𝐻𝑔 -1 . By lemma 1.2.10, since 𝐻 is compact, 𝑔 belongs to The action of 𝐺 on 𝑉 induces an action of 𝑁 (𝐻) on 𝑉 𝐻 and we can therefore reduce (locally) the problem of describing the orbit space of 𝑉 /𝐺 by the orbit space of 𝑉 𝐻 /𝑁 (𝐻). Lemma 1.2.25. Given an isotropy subgroup 𝐻 and a vector 𝑣 𝑣 𝑣 ∈ 𝑉 𝐻 then, for all 𝑤 𝑤 𝑤 ∈ Σ [𝐻] ∩𝑉 𝐻 , 𝑤 𝑤 𝑤 belongs to the 𝐺-orbit of 𝑣 𝑣 𝑣 in 𝑉 ⇐⇒ 𝑤 𝑤 𝑤 belongs to the 𝑁 (𝐻)-orbit of 𝑣 𝑣 𝑣 in 𝑉 𝐻 . Proof. Let 𝑣 𝑣 𝑣 ∈ 𝑉 𝐻 and 𝑤 𝑤 𝑤 ∈ Σ [𝐻] ∩ 𝑉 𝐻 such that 𝑤 𝑤 𝑤 ∈ 𝐺 • 𝑣 𝑣 𝑣 then there exists 𝑔 ∈ 𝐺 such that 𝑤 𝑤 𝑤 = 𝑔 • 𝑣 𝑣 𝑣. Since 𝑤 𝑤 𝑤 ∈ Σ [𝐻] ∩ 𝑉 𝐻 and by lemma 1.2.10, 𝐻 = 𝐺 𝑤 𝑤 𝑤 . Hence, 𝐻 = 𝑔𝐺 𝑣 𝑣 𝑣 𝑔 -1 and we have 𝐻 ⊂ 𝐺 𝑣 𝑣 𝑣 (since 𝑣 𝑣 𝑣 ∈ 𝑉 𝐻 ). Applying one more time lemma 1.2.10, we deduce that 𝐺 𝑣 𝑣 𝑣 = 𝑔𝐺 𝑣 𝑣 𝑣 𝑔 -1 = 𝐻. Therefore, 𝑔 ∈ 𝑁 (𝐻) and hence 𝑤 𝑤 𝑤 ∈ 𝑁 (𝐻) • 𝑣 𝑣 𝑣. The converse is straightforward.The representation 𝜌 𝑁 (𝐻) of 𝑁 (𝐻) on 𝑉 𝐻 is not faithful. When 𝐻 is an isotropy group, 𝑉 𝐻 is called a linear slice and Γ will be its monodromy group. . In particular, when Γ is the trivial group then each 𝐺-orbit meets 𝑉 𝐻 in at most one point.

	𝑁 (𝐻).
	its kernel applying lemma 1.1.6, we get the following faithful linear representation ( ker(𝜌 𝑁 (𝐻) ) = { }) is exactly 𝐻 therefore, by 𝑔 ∈ 𝑁 (𝐻) | ∀𝑣 𝑣 𝑣 ∈ 𝑉 𝐻 , 𝜌 𝑁 (𝐻) (𝑔)𝑣 𝑣 𝑣 = 𝑣 𝑣 𝑣
		𝜌 Γ : Γ -→ GL(𝑉 𝐻 ) where Γ := 𝑁 (𝐻)/𝐻.
	When Γ is finite, each 𝐺-orbit intersects 𝑉 𝐻 at most in a finite number of points, in this case,
	1.3	Algebras of invariants and covariants
	1.3.1 Action of a group on the algebra of invariants
	Let K be the field R or C and consider the polynomial algebra K[𝑉 ] consisting of all polynomial
	functions on 𝑉 with coefficients in K. The linear action of the group 𝐺 on the vector space 𝑉
	extends naturally to K[𝑉 ] setting as follows

1

A

  𝐺 = K[𝐼 1 , . . . , 𝐼 𝑁 ].Such a family 𝐼 1 , . . . , 𝐼 𝑁 is called an integrity basis.

	Theorem 1.3.1. For every linear algebraic reductive group, the invariant algebra K[𝑉 ] 𝐺 is
	finitely generated.
	This means that there exists a finite set of invariant polynomials 𝐼 1 , . . . , 𝐼 𝑁 which generate
	the invariant algebra K[𝑉 ] 𝐺 :
	K[𝑉 ] The actual determination of an integrity basis remains a very active field of research. Below,
	we give some examples.
	Example 1.3.2. 1. The case of the natural representation (SO(3), R 3 ) for which an integrity
	basis is simply given by the squared norm:

  2 , 𝐼 3 ], 𝐼 𝑘 := tr(a 𝑘 ), 𝑘 = 1, 2, 3.3. The case of standard representation of the group of permutations 𝒮 𝑛 on R 𝑛 : R[R 𝑛 ] 𝒮𝑛 = R[𝜎 1 , . . . , 𝜎 𝑛 ] where 𝜎 𝑖 are the elementary symmetric functions 𝜎 1 (𝑥 𝑥 𝑥) = 𝑥 1 + . . . + 𝑥 𝑛 , 𝜎 2 (𝑥 𝑥 𝑥) = Σ 𝑖̸ =𝑗 𝑥 𝑖 𝑥 𝑗 , . . . , 𝜎 𝑛 (𝑥 𝑥 𝑥) = 𝑥 1 . . . 𝑥 𝑛 , 𝑥 𝑥 𝑥 = (𝑥 1 , ; . . . , 𝑥 𝑛 ) ∈ R 𝑛 . , 𝑣 𝑣 𝑣 2 ∈ 𝑉 belonging to different orbits, it is always possible to find a function 𝑝 ∈ R[𝑉 ] 𝐺 such that 𝑝(𝑣 1 ) ̸ = 𝑝(𝑣 2 ) (see for instance [1, Appendix C]). In other words, ∀𝑝 ∈ R[𝑉 ] 𝐺 , 𝑝(𝑣 𝑣 𝑣 1 ) = 𝑝(𝑣 𝑣 𝑣 2 ) ⇐⇒ 𝑣 𝑣 𝑣 1 = 𝑔 • 𝑣 𝑣 𝑣 2 for some 𝑔 ∈ 𝐺.Example 1.3.5. For instance, two vectors 𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤 ∈ R 2 are related by a rotation if and only if they have the same norms ‖𝑣 𝑣 𝑣‖ 2 = ‖𝑤 𝑤 𝑤‖ 2 .

	Definition 1.3.3. A minimal integrity basis is an integrity basis such that none of its elements
	can be expressed as a polynomial of the others.
	Remark 1.3.4. A minimal integrity basis is not unique, however, all of them have the same
	cardinal and the list of the degrees of the generators are the same (see for instance [44, section
	5.1.1]).
	An important property of polynomial invariants for a real representation of a compact group
	(and thus of any integrity basis) is that they separate the orbits, which means that given two
	vectors 𝑣 𝑣 𝑣 1

  𝑝 𝑖𝑗𝑘 (𝑣 𝑣 𝑣)𝑥 𝑖 𝑦 𝑗 𝑧 𝑘 , where each coefficient 𝑝 𝑖𝑗𝑘 (𝑣 𝑣 𝑣) is a polynomial function of 𝑣 𝑣 𝑣, and such that p(𝑔 • 𝑣 𝑣 𝑣, 𝑥 𝑥 𝑥) = p(𝑣 𝑣 𝑣, 𝑔 -1 • 𝑥 𝑥 𝑥), for all 𝑣 𝑣 𝑣 ∈ V, 𝑥 𝑥 𝑥 ∈ R 3 and 𝑔 ∈ SO(3).

	is thus a polynomial which can be written as
	p(𝑣 𝑣 𝑣, 𝑥 𝑥 𝑥) =	∑
		𝑖,𝑗,𝑘
	Like the invariant algebra, the covariant algebra of a finite-dimensional representation of a
	compact group is finitely generated.
	Theorem 1.3.10 (Hilbert's Theorem [50]). The covariant algebra Cov(V) is finitely generated,
	i.e. there exists a finite set ℬ := {p 1 , . . . , p 𝑠 } in Cov(V) such that

  • If 𝑠 1 , 𝑠 2 are two points in 𝑆 and there exists 𝑔 in 𝐺 with 𝑔 • 𝑠 1 = 𝑠 2 , then 𝑔 belongs to 𝐺 𝑣 . In particular, 𝐺 𝑠 ⊂ 𝐺 𝑣 for each point 𝑠 ∈ 𝑆. • 𝐺 • 𝑆 is an open neighborhood of the orbit 𝐺 • 𝑣. Given a slice 𝑆, the open 𝐺-stable neighborhood 𝐺 • 𝑆 is diffeomorphic to the quotient 𝐺 × 𝐺𝑣 𝑆, and after linearizing the action of 𝐺 𝑣 on 𝑆, 𝐺 • 𝑆 is locally diffeomorphic to 𝐺 × 𝐺𝑣 𝑁 𝑣 where 𝑁 𝑣 is the normal space of the orbit at 𝑣 (viewed as a representation of the isotropy group 𝐺 𝑣 ). In our settings, where 𝐺 is compact, the two last properties are automatic (of course up to shrinking 𝑆) as soon as the two first ones are granted. • The last property can be made slightly stronger: up to shrinking 𝑆, we can even assume that the orbit map from 𝐺 × 𝑆 to 𝑉 is a submersion, in particular it is an open map. Let 𝑣 in 𝑉 . Then there exists a neighborhood 𝑈 of 𝑣 such that for all 𝑤 in 𝑈 , [𝐺 𝑤 ] ⪯ [𝐺 𝑣 ]. Proof. Let 𝑆 be a slice at 𝑣. If 𝑤 ∈ 𝐺 • 𝑆 then 𝑤 is in the orbit of a point 𝑤 ′ in 𝑆. It follows that 𝐺 𝑤 ′ ⊂ 𝐺 𝑣 , so [𝐺 𝑤 ] = [𝐺 𝑤 ′ ] ⪯ [𝐺 𝑣 ]. □ Definition 2.4. A vector 𝑣 ∈ 𝑉 (or its orbit) is called principal if there exists a neighborhood 𝑈 of 𝑣 such that for all 𝑤 in 𝑈 , [𝐺 𝑤 ] = [𝐺 𝑣 ]. Equivalently, 𝑣 is principal if it has locally minimal isotropy. The definition of principal points is not totally standard. Some authors, like [1, section 6] have adopted another definition: A vector 𝑣 ∈ 𝑉 is principal if there exists a neighborhood 𝑈 of 𝑣 such that for all 𝑤 in 𝑈 , [𝐺 𝑣 ] ⪯ [𝐺 𝑤 ]. Anyway, for a compact Lie group representation, the two definitions are equivalent. For any open subset Ω of 𝑉 such that 𝜋(Ω) is connected, the set 𝑈 Ω of principal points in Ω is open, dense in Ω, and 𝜋(𝑈 Ω ) is connected. Let 𝐺 be a compact Lie group acting linearly on 𝑉 and 𝐻 be an isotropy subgroup.• The fixed locus 𝑉 𝐻 is defined by 𝑉 𝐻 := {𝑣 ∈ 𝑉 ; ℎ • 𝑣 = 𝑣 for all ℎ ∈ 𝐻} , it is the vector subspace of elements of 𝑉 fixed by 𝐻.

	• Let us denote by 𝜋 the canonical projection from 𝑉 to 𝑉 /𝐺. The principal orbit type theorem
	can be stated (in the case of linear representations) as follows (see [7, Theorem 3.1], [1, section
	VI], [10, Theorem 2.8.5]).
	Theorem 2.6 (Principal orbit type theorem). For the interested reader, we present a proof in Appendix B. It follows from this result that
	there is a unique minimal orbit type.
	Definition 2.7. • The open fixed locus
	It is the set of vectors in 𝑉 having isotropy class [𝐻] and it is the orbit of	∘ 𝑉 𝐻 .
	Remark 2.2. We have furthermore the following properties. • The closed stratum Σ [𝐻] = { 𝑣 ∈ 𝑉 ; 𝑔𝐻𝑔 -1 ⊂ 𝐺 𝑣 , for some 𝑔 ∈ 𝐺 consists of vectors 𝑉 having isotropy at least [𝐻]. It is the orbit of 𝑉 𝐻 . } Remark 2.8. • • Slices exist in general for proper actions of Lie groups on manifolds [10, Theorem 2.3.1], and in particular for actions of compact Lie groups. However, for linear representations of compact Lie groups, it is possible to produce slices easily (see lemma B.3 below). • The terminology we use here is not totally standard, but it varies according to different authors (for instance, what we call stratum is called orbit bundle in
	• The existence of slices has the following corollary.
	Corollary 2.3. Remark 2.5.
	• By definition, the set of principal points is open.

∘

𝑉 𝐻 is the subset of 𝑉 𝐻 defined by

∘ 𝑉 𝐻 := { 𝑣 ∈ 𝑉 𝐻 ; 𝐺 𝑣 = 𝐻 } . • The stratum Σ [𝐻] is defined by Σ [𝐻] := {𝑣 ∈ 𝑉 ; [𝐺 𝑣 ] = [𝐻]} .

  and 𝑔 ∈ 𝐺 is such that 𝑔 • 𝑉 𝐻 ⊂ 𝑉 𝐻 , then, 𝑔 • 𝑣 0 is in 𝑉 𝐻 and thus 𝐻 ⊂ 𝑔𝐻𝑔 -1 , because 𝐺 𝑔•𝑣 0 = 𝑔𝐺 𝑣 0 𝑔 -1 . Since 𝐻 is compact, 𝐻 = 𝑔𝐻𝑔 -1 by[38, Proposition 3.7] and 𝑔 belongs to 𝑁 (𝐻), which ends the proof.□ 𝐻 := 𝑁 (𝐻)/𝐻 is called the monodromy group in[2]. Note that, in that case, two vectors𝑣 1 , 𝑣 2 in 𝑉 𝐻 ∩ Σ [𝐻]are in the same 𝐺-orbit if and only if they are in the same Γ 𝐻 -orbit. 𝑉 𝐻 is open and dense in 𝑉 𝐻 , more precisely it contains a Zariski dense open subset of 𝑉 𝐻 . 𝑉 𝐻 . Then thanks to corollary 2.3 there exists a neighborhood 𝑈 of 𝑥 such that for 𝑦 in 𝑈 , [𝐺 𝑦] ⪯ [𝐻]. If 𝑦 in 𝑈 ∩ 𝑉 𝐻 , [𝐻] ⪯ [𝐺 𝑦 ] ⪯ [𝐻] so [𝐺 𝑦 ] = [𝐻]. This means that 𝐻 ⊂ 𝐺 𝑦 = 𝑔𝐻𝑔 -1 for some 𝑔 in 𝐺 so by[38, Proposition 3.7] again, 𝐺 𝑦 = 𝐻. This proves that𝑉 𝐻 is open. The second part is difficult and will be proven later on (see corollary3.17).□ Corollary 2.11. Σ [𝐻] is the closure of Σ [𝐻] . Proof. First we prove that Σ [𝐻] is closed. If 𝑣 𝑛 is a sequence in Σ [𝐻]that converges to 𝑣 then there exists elements 𝑔 𝑛 in 𝐺 such that 𝑔 𝑛 𝐻𝑔 -1 𝑛 ⊂ 𝐺 𝑣𝑛 . After extracting a subsequence, we can assume that 𝑔 𝑛 converges to an element 𝑔 in 𝐺. Then 𝑔𝐻𝑔 -1 ⊂ 𝐺 𝑣 so 𝑣 belongs to Σ[𝐻] . Now, let 𝑣 be in Σ[𝐻] . By definition, there exists 𝑔 in 𝐺 such that 𝑔 -1 • 𝑣 belongs to 𝑉 𝐻 . Thanks to proposition 2.10, 𝑔 -1 • 𝑣 is the limit of a sequence of elements in

	The linear representation 𝜌 : 𝐺 → GL(𝑉 ) induces, by restriction, a linear representation of 𝑁 (𝐻) on 𝑉 𝐻
	𝜌 𝑁 (𝐻) : 𝑁 (𝐻) -→ GL(𝑉 𝐻 ). This induced representation is however not faithful in general but, when 𝐻 is an isotropy group,
	its kernel is exactly 𝐻 and we get a faithful linear representation
	𝜌 Γ 𝐻 : Γ 𝐻 -→ GL(𝑉 𝐻 ),
	where Γ Proposition 2.10. Let 𝐻 be an isotropy subgroup. Then
	Proof. Let 𝑥 be in
	∘ 𝑉 𝐻 .	□
	For a compact Lie group representation (𝑉, 𝜌), the partition into (non empty) isotropy strata

∘ ∘ ∘

  Proposition 3.16. For any isotropy 𝐻, the complex open fixed locus in (𝑉 C ) 𝐻 C contains a nonempty Zariski open set. Proof. Let 𝑣 be a vector in 𝑉 such that 𝐺 𝑣 = 𝐻. Thanks to proposition 3.14, 𝐺 C • 𝑣 is closed, and thanks to corollary 2.20, there exists a Zariski open set 𝑈 containing 𝑥 such that for any 𝑤 in 𝑈 , 𝐺 𝑤 is conjugate to a subgroup of 𝐺 𝑣 . Let us now look at ̃ 𝑈

  Then the result follows from corollary 3.8. □ Corollary 3.22. If {𝐽 1 , . . . , 𝐽 𝑁 } is a set of generators for R[𝑉 ] 𝐺 , then, it is also a generating set for C[𝑉 C ] 𝐺 C . Lemma 3.23. Let 𝐻 be a closed subgroup of a Lie group 𝐺 and let Γ be a set of representatives in 𝐻 of the finite group 𝐻/𝐻 𝑒 , where 𝐻 𝑒 is the identity component of 𝐻. Then,

3.3. Complexification of normalizers. The aim of this section is to prove that the complexification of the normalizer of a compact subgroup of a compact Lie group is naturally isomorphic to the normalizer of the corresponding complexified subgroup (proposition 3.26). This result is in accordance with Luna's result

[30, Lemma 1.1]

. In the following three lemmas, 𝐺 denotes a compact Lie group acting on a vector space 𝑉 and 𝐻 is a Lie subgroup of 𝐺. We denote by 𝐺 C and 𝐻 C their respective complexifications.

  7, 𝐶 𝑔 vanishes on 𝐻 C . The same argument applies for 𝑔 -1 so 𝑔 normalizes 𝐻 C in 𝐺 C . Hence 𝑁 𝐺 (𝐻) ⊂ 𝑁 𝐺 C (𝐻 C ) and by theorem 3.3 we get (𝑁 𝐺 (𝐻)) C ⊂ 𝑁 𝐺 C (𝐻

  1. Let {𝐽 1 , . . . , 𝐽 𝑁 } be a generating set for the invariant algebra R[𝑉 ] 𝐺 . Then {𝐽 1 , . . . , 𝐽 𝑁 } separates the orbits of (𝑉, 𝐺). Let 𝑗 𝑘 be the restriction of 𝐽 𝑘 on 𝑉 𝐻 , then, the 𝑗 𝑘 separate the orbits of 𝑉 𝐻 ∩ Σ [𝐻] . thanks to proposition 3.12, there exists a nonempty Zariski open subset 𝑈 2 of 𝑊 such that all 𝐾 C -orbits of elements of 𝑈 2 are closed in 𝑊 . Let 𝑈 = 𝑈 1 ∩ 𝑈 2 , then, 𝑈 is also a nonempty Zariski open set in 𝑊 .We claim that {𝑗 1 , . . . , 𝑗 𝑁 } separates 𝐾 C -orbits of elements in 𝑈 . To prove this, let 𝑤 1 and 𝑤 2 be two elements in 𝑈 that are not in the same 𝐾 C -orbit. Assume that 𝑤 2 and 𝑤 1 are in the same 𝐺 C -orbit. If 𝑔𝑤 1 = 𝑤 2 for some 𝑔 in 𝐺 C , then 𝐻 C = 𝑔𝐻 C 𝑔 -1 and 𝑔 belongs to 𝑁 𝐺 C (𝐻 C ). Thanks to proposition 3.26, 𝑔 is in 𝐾 C , which gives a contradiction. Hence 𝐺 C 𝑤 1 ∩ 𝐺 C 𝑤 2 = ∅.The 𝐾 C -orbits of 𝑤 1 and 𝑤 2 are closed because 𝑤 1 and 𝑤 2 belong to 𝑈 2 . Besides, thanks to proposition 3.26, 𝐾 C = 𝑁 𝐺 C (𝐻 C ). It follows from [29, Corollary 1] that the 𝐺 C -orbits of 𝑤 1 and 𝑤 2 are also closed. Since these orbits are constructible, they are Zariski closed (and also 𝐺 C -stable). Thanks to corollary 3.22, they can be separated by the 𝐽 𝑘 . This proves our claim. By Popov & Vinberg's theorem 2.15, the 𝑗 𝑘 generate the field (C(𝑊 )) 𝐾 C . It follows that the map (C(𝑉 C )) 𝐺 C (C(𝑊 )) 𝐾 C is surjective. It remains to go back to real fields of invariants. Thanks to corollary 3.8, we have a diagram

Let 𝑊 = (𝑉 C ) 𝐻 C . Thanks to proposition 3.16, there exists a nonempty Zariski open set 𝑈 1 of 𝑊 such that for any 𝑤 in 𝑈 1 , 𝐺 C 𝑤 = 𝐻 C . Let 𝐾 = 𝑁 𝐺 (𝐻). The group 𝐾 acts on 𝑊 , and

  16, there exists a Zariski subset 𝑈 in (𝑉 C ) 𝐻 C which is contained in the open fixed locus. Let 𝑋 denotes the 𝐺 C orbit of 𝑈 in 𝑉 C , then by Chevalley's theorem 𝑋 is constructible in 𝑉 C , so its topological closure is the same as its Zariski closure. 𝑍 is in the Lie algebra of 𝑁 𝐺 (𝐺 𝑣 ) so 𝑒 𝑖𝑍 normalizes 𝐺 C 𝑣 . It follows that𝐻 C = 𝑔𝑒 𝑖𝑍 𝐺 C 𝑣 𝑒 -𝑖𝑍 𝑔 -1 = 𝑔 𝐺 C 𝑣 𝑔 -1 .Taking the fixed points under the Cartan involution on both sides, 𝐻 = 𝑔 𝐺 𝑣 𝑔 -1 , which implies that 𝑣 lies in the 𝐺-orbit of Let us now prove that 𝑋 ∩ 𝑉 = Σ[𝐻] . The inclusion follows by taking the closure of the inclusion 𝑋 ∩ 𝑉 ⊂ Σ 𝐻 . For the other inclusion, recall that 𝑈 is a Zariski open subset of 𝑊 and that 𝑋 is the 𝐺 C -orbit or 𝑈 . We have thus 𝑈 ∩ 𝑉 𝐻 ⊂ 𝑋 ∩ 𝑉 and thanks to corollary A.8, 𝑈 ∩ 𝑉 𝐻 is a nonempty Zariski subset of 𝑉 𝐻 . Hence, 𝑉 𝐻 ⊂ 𝑋 ∩ 𝑉 . Since the right hand side is 𝐺-stable, we get Σ [𝐻] ⊂ 𝑋 ∩ 𝑉 .We can finish the proof as follows: 𝑋 is Zariski open in 𝑋, so 𝑋 ∩ 𝑉 is Zariski open in the algebraic set 𝑋 ∩ 𝑉 . Therefore, 𝑋 ∩ 𝑉 is also an algebraic set, which is the union of a finite number of irreducible components of 𝑋 ∩ 𝑉 . □

	Remark 4.1.

Let us first prove that 𝑋 ∩ 𝑉 lies in Σ 𝐻 . If 𝑣 is in 𝑋 ∩ 𝑉 , then there exists ℓ in 𝐺 C such that ℓ • 𝑣 belongs to 𝑈 . This implies 𝐺 C ℓ•𝑣 = 𝐻 C , and therefore ℓ 𝐺 C 𝑣 ℓ -1 = 𝐻 C . Since 𝐺 C 𝑣 and 𝐻 C are stable under the Cartan involution 𝜄, we also have 𝜄(ℓ) 𝐺 C 𝑣 𝜄(ℓ) -1 = 𝐻 C . It follows that 𝜄(ℓ) -1 .ℓ normalizes 𝐺 C 𝑣 . If ℓ = 𝑔𝑒 𝑖𝑍 , then 𝜄(ℓ) -1 .ℓ = 𝑒 2𝑖𝑍 and thanks to proposition 3.26, ∘ 𝑉 𝐻 . • In the first part of the proof, we made crucial use of Luna's criterion for closedness of orbits [29, Corollary 1]

. For an easy proof of this theorem using Kempf-Ness theory, see

[26]

.

• The natural candidate for Σ [𝐻] should be 𝑋 ∩ 𝑉 . However, we don't know if 𝑋 ∩ 𝑉 is irreducible in full generality. Hence the best result we could get is that Σ [𝐻] is one of its irreducible components.

Appendix A. Lie groups and real analytic structures

  • For any 𝑔 in 𝐺 the map 𝐿 𝑔 ∘ exp in a neighborhood of 0 defines a local real analytic chart near 𝑔. Besides, if 𝐻 is a Lie subgroup of 𝐺, then 𝐻 is a real analytic submanifold of 𝐺. More generally, any continuous morphism between compact Lie groups is real analytic. Lastly, let us state the following lemma, which is the differentiable version of [38, Proposition 3.7].

  B.1. Let 𝑋 be a connected topological space, 𝑈 a dense subset of 𝑋, and assume than for any point 𝑥 in 𝑋, there exists a neighborhood 𝑈 𝑥 of 𝑥 such that 𝑈 ∩ 𝑈 𝑥 is connected. Then 𝑈 is connected.Remark B.2. This lemma can be better visualized in term of sheaf theory, let 𝑗 : 𝑈 ˓→ 𝑋 denote the inclusion, and let us consider the morphism of sheaves ∆ : Z 𝑋 → 𝑗 * Z 𝑈 . Then the two hypotheses can be translated as follows:• 𝑈 is dense in 𝑋 if and only if ∆ is a monomorphism.

Proof. Assume that 𝑈 can we written as the disjoint union of two nonempty open sets 𝑈 1 and 𝑈 2 . Since 𝑈 is dense, 𝑋 = 𝑈 1 ∪ 𝑈 2 and since 𝑋 is connected,

𝑈 1 ∩ 𝑈 2 ̸ = ∅. Let 𝑥 in 𝑈 1 ∩ 𝑈 2 .

Let 𝑈 𝑥 be a neighborhood of 𝑥 such that 𝑈 ∩ 𝑈 𝑥 is connected. Then 𝑈 ∩ 𝑈 𝑥 is the disjoint union of 𝑈 1 ∩ 𝑈 𝑥 and 𝑈 2 ∩ 𝑈 𝑥 . However these two open sets are non empty, and we get a contradiction. Hence 𝑈 is connected. □

  therefore 𝑈 is connected. Now we explain how to construct linear slices explicitly. Lemma B.3. Assume that 𝐺 acts by orthogonal transformations on 𝑉 . For any 𝑣 in 𝑉 , let 𝐸 𝑣 be the tangent space of the orbit 𝐺 • 𝑣 at 𝑣. Then, 𝐸 ⊥ 𝑣 is a linear subspace of 𝑉 , stable by 𝐺 𝑣 , and meets all 𝐺-orbits in 𝑉 . Besides, a neighborhood of 𝑣 in 𝐸 ⊥ 𝑣 is a local slice of the 𝐺-action at 𝑣.

  3, we can apply the induction hypothesis with the action of 𝐺 𝑣 on 𝐸 ⊥ 𝑣 which has smaller dimension. Let us consider a small 𝐺 𝑣 -stable open ball 𝐵 centered at 𝑣 in 𝐸 ⊥ 𝑣 . If the radius is small enough, 𝐵 is a local slice of the 𝐺-action at 𝑣. Let 𝒫 be the set of principal points in 𝐵. By induction, since 𝐵 is open and connected in 𝐸 ⊥ 𝑣 , 𝒫 is open and dense in 𝐵, and 𝒫/𝐺 𝑣 is connected. Let us now remark that since 𝐵 is a local slice, for any 𝑣 ′ in 𝐵, (𝐺 𝑣 ) 𝑣 ′ = 𝐺 ′ 𝑣 . It follows that 𝒫 consists of principal points of 𝐵, but also of 𝑉 . This proves that 𝑣 is in the closure of 𝑈 Ω , so 𝑈 Ω is dense in Ω. Since 𝐵 is a local slice, we can assume, up to shrinking 𝐵, that the action map 𝐺 × 𝐵 → 𝑉 is an open map. Hence, 𝑈 𝑣 = 𝐺.𝒫 is an open subset of principal points in 𝑉 , and furthermore 𝑈 𝑣 /𝐺 = 𝒫/𝐺 𝑣 is connected.

Table 3 .

 3

1: Generators of closed O(3)-subgroups and D 𝑛 by

  3 ) is irreducible for the SO(3)-representation 𝜌 𝑛 . Moreover, every irreducible representation of SO(3) is equivalent to (H 𝑛 (R 3 ), 𝜌 𝑛 ) for some integer 𝑛.Remark 3.2.4. Two representations (𝑉, 𝜌 1 ) and (𝑊, 𝜌 2 ) of a group 𝐺 are said to be equivalent if there exists an equivariant (definition 1.1.9) linear bijective map from 𝑉 into 𝑊 (see[14, definition 1.4]).Corollary 3.2.5. (Harmonic decomposition) Every finite dimensional representation 𝑉 of the rotation group SO(3) splits into a direct sum of irreducible representations, each of them being isomorphic to an harmonic tensor space H 𝑛 (R 3 ).

	As

a consequence, we can deduce a similar result for O(3)-representation. Denote by H #𝑛 (R 3 ) the space of harmonic tensors of order 𝑛 endowed with the twisted representation ρ𝑛 defined in (3.2). Corollary 3.2.6. [85, corollary 3.1.2] The subspaces H 𝑛 (R 3 ) and H #𝑛 (R 3 ) are irreducible for the O(3)-representation. Moreover, every irreducible representation of O(3) is equivalent to

  2 3 together with the hypothesis equation 196𝐽 6 -51𝐽 3 2 + 216𝐽 2 𝐽 4 + 10𝐽 2 3 = 0 will give the following equation 5𝐽 3 2 -8𝐽 2 𝐽 4 -70𝐽 2 3 = 0 which contradicts the fact that H ∈ Σ [D 4 ] .

5. Using [91, theorem 9.10, corollary 8.4] we deduce the first implication.

  linear representations of a group 𝐺, if we know the isotropy classes for each individual representation. Such an operation has been generalized by Marc Olive[START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF] to all conjugacy classes of closed subgroups of a given group 𝐺 and is defined as follows Definition 3.3.1. For two subgroups 𝐻 1 and 𝐻 2 of a group 𝐺, we define the clips operation of the conjugacy classes [𝐻 1 ] and [𝐻 2 ] as the set of conjugacy classes

Table 2 ]

 2 I, type II or type III. The clips tables have already been established for two type I subgroups (see[18, table 1],[START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF] Table 1] and subsection 3.4.1) and for two type III subgroups (see[86, ). The clips operation between a type I and a type III subgroup is deduced from[START_REF] Olive | Effective computation of SO (3) and O(3) linear representation symmetry classes[END_REF] Lemma 5.4]. The clips between a type I and a type II subgroup or two type II subgroups are deduced from the clips between two type I subgroups, see remark 3.4.1 below. However, the clips between type II and type III subgroups do not seem to have already been calculated. The main result of this chapter is their calculation, the results are summarized in tables 3.2 and 3.3 of the next section. Remark 3.4.1. Notice that 1. if 𝐻 1 and 𝐻 2 are two closed subgroups of SO(3), then,

table 1

 1

] and

[86,

Table 1

]. However, these two tables differ in some cases. In the following remark, we point out the differences between the two references and we recalculate the disputed clips. Remark 3.4.2. The two tables regrouping the clips between SO(3)-subgroups provided in

[18,

table 1] and [86, Table

1

] differ in the following cases:

Clips between [D 𝑛 ] and [O(2)] (see equations

(3.10)

and

  We get solutions only if 𝑚 1 = 𝑚 𝑑 is even. By replacing 𝑚 1 by 2𝑝 we get that 𝑝 divides 𝑘 2 hence 𝑘 2 = 𝑝𝑘 ′ . Replacing 𝑚 1 by 2𝑝 and 𝑘 2 by 𝑝𝑘 ′ in the equation

.

19

)

If 𝑚 = 𝑚 1 𝑑 and 𝑛 = 𝑛 1 𝑑, for 𝑚 1 , 𝑛 1 ∈ N, then the equation (3.19) becomes (2𝑘 1 + 1)𝑑𝑚 1 = 2𝑘 2 𝑑𝑛 1 with gcd(𝑚 1 , 𝑛 1 ) = 1.

  𝑒 𝑖 ] for a rotation 𝑔 such that, , ∃𝑘 = 1, . . . , 6 and ∃𝑖 = 1, . . . , 𝑛, 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = b 𝑖 (for instance take 𝑔 = r ( 𝑒 𝑒 𝑒 1 , 𝜋 𝑎 𝑎 𝑎 𝑐 1 to 𝑒 𝑒 𝑒 1 ), [Z 𝑑 2 ] for a rotation 𝑔 such that, ∃𝑘 = 1, . . . , 6, 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = 𝑒 𝑒 𝑒 3 , [Z 𝑑 3 ] for a rotation 𝑔 such that, ∃𝑗 = 1, . . . , 4, 𝑔𝑠 𝑠 𝑠 𝑡 𝑗 = 𝑒 𝑒 𝑒 3 , [Z 𝑑 4 ] for a rotation 𝑔 such that, ∃𝑖 = 1, . . . , 3, 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 3 (take for instance 𝑔 = r 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = 𝑒 𝑒 𝑒 3 and two other edge axes 𝑎 𝑎 𝑎 𝑐 𝑘 turn to two orthogonal axes ±b 𝑖 for 𝑖 = 1, . . . , 𝑛 (such 𝑔 exists for 𝑛 even), [D 𝑧 𝑑 3 ] for a rotation 𝑔 such that 𝑔𝑠 𝑠 𝑠 𝑡 𝑗 = 𝑒 𝑒 𝑒 3 and three other edge axes 𝑎 𝑎 𝑎 𝑐 𝑘 (for instance 𝑎 𝑎 𝑎 𝑐 1 , 𝑎 𝑎 𝑎 𝑐 4 and 𝑎 𝑎 𝑎

	4	⋃ 4 𝑗=1 Z 𝑠 𝑠 𝑠𝑡 𝑗 3	⋃ 6 𝑘=1 Z 𝑎 𝑎 𝑎𝑐 𝑘 2	(3.12), we get
	[Z -2 3 r ( 𝑒 𝑒 𝑒 1 , 𝜋 2 ) ),	)	∘ r	( 𝑒 𝑒 𝑒 3 , -𝜋 4	)	that turns only ( 𝑒 𝑒 𝑒 3 , 𝜋 3	)	∘
	[D 𝑧 𝑑

2 ] for a rotation 𝑔 such that, ∃𝑖 = 1, . . . , 6,

  𝑢 𝑖 𝑑 2 ] for a rotation 𝑔 such that, ∃𝑘 = 1, . . . , 15, 𝑔𝑤 𝑤 𝑤 𝑘 = 𝑒 𝑒 𝑒 3 , [Z 𝑑 3 ] for a rotation 𝑔 such that, ∃𝑗 = 1, . . . , 10, 𝑔𝑣 𝑣 𝑣 𝑗 = 𝑒 𝑒 𝑒 3 , [Z 𝑑 5 ] for a rotation 𝑔 such that, ∃𝑖 = 1, . . . , 6, 𝑔𝑢 𝑢 𝑢 𝑖 = 𝑒 𝑒 𝑒 3 , [D 𝑧 𝑑 2 ] for a rotation 𝑔 such that 𝑔𝑤 𝑤 𝑤 𝑘 = 𝑒 𝑒 𝑒 3 and two other axes 𝑤 𝑤 𝑤 𝑘 turn to two orthogonal axes b 𝑖 for 𝑖 = 1, . . . , 𝑛 (such a 𝑔 exists for 𝑛 even) (the identity rotation works as well since 𝑤 𝑤 𝑤 4 , 𝑤 𝑤 𝑤 10 , 𝑤 𝑤 𝑤 12 are colinear to 𝑒 𝑒 𝑒 1 , 𝑒 𝑒 𝑒 2 , 𝑒 𝑒 𝑒 3 (𝑒 𝑒 𝑒 2 = b 𝑖 for some 𝑖 for 𝑛 even)), [D 𝑧 𝑑 3 ] for a rotation 𝑔 such that 𝑔𝑣 𝑣 𝑣 𝑗 = 𝑒 𝑒 𝑒 3 and three other axes 𝑤 𝑤 𝑤 𝑗 turn to three b 𝑖 . However, there is no three coplanar 𝑤 𝑤 𝑤 𝑘 separated by an angle of 𝜋 3 , [D 𝑧 𝑑 5 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 3 and five axes 𝑤 𝑤 𝑤 𝑖 turn to five b 𝑖 . However, there is no five coplanar 𝑤 𝑤 𝑤 𝑘 separated by angle of 𝜋 5 .

	5	⋃ 10 𝑗=1 Z 𝑣 𝑣 𝑣 𝑗 3	⋃ 15 𝑘=1 Z 𝑤 𝑤 𝑤 𝑘

2 (3.14), we get [Z - 2 ] for a rotation 𝑔 such that, ∃𝑖 = 1, . . . , 𝑛 and ∃𝑘 = 1, . . . , 15, 𝑔𝑤 𝑤 𝑤 𝑘 = b 𝑖 , [Z

  ±𝑒 𝑒 𝑒 𝑖 and we get [Z - 4 ] if 4 | 𝑚 and [Z 𝑑 ′ 2 ] if not, 𝑔𝑒 𝑒 𝑒 3 = ±𝑠 𝑠 𝑠 𝑡 𝑗 and we get [Z 𝑑 ′ 3 ], 𝑔𝑒 𝑒 𝑒 3 = ±𝑎 𝑎 𝑎 𝑐 𝑘 and we get [Z -

	𝑑 ′ 2

  for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑠 𝑠 𝑠 𝑡 𝑗 , ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑠 𝑠 𝑠 𝑡 𝑗 and three secondary axis b 𝑖 of D 𝑚 turn to three 𝑎 𝑎 𝑎 𝑐 𝑘 , [D 2 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑒 𝑒 𝑒 𝑖 and two orthogonal secondary axis of D 𝑚 turn to the two remaining 𝑒 𝑒 𝑒 𝑖 and that is possible when 𝑚 is even and 4 ∤ 𝑚 since if 4 | 𝑚 we get 𝑧 2 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑒 𝑒 𝑒 𝑖 and two orthogonal secondary axis of D 𝑚 turn to orthogonal 𝑎 𝑎 𝑎 𝑐 𝑘 and this is possible if 𝑚 is even, [Z - 4 ] for a rotation 𝑔 such that 𝑔𝑒 𝑒 𝑒 3 = 𝑒 𝑒 𝑒 𝑖 for 𝑖 = 1, . . . , 3 and if 4 | 𝑚.

	[D 𝑧 𝑑 ′ 3
	[D 𝑑 4 ],
	[D

  except [D 2 ] since to get [D 2 ] we need a rotation 𝑔 such that 𝑔𝑎 𝑎 𝑎 𝑐 𝑘 = 𝑒 𝑒 𝑒 𝑖 and 𝑔𝑒 𝑒 𝑒 𝑖 = 𝑒 𝑒 𝑒 𝑖 which will give [Z -

4

]. We have

  𝐸 1111 𝐸 1122 𝐸 1133 𝐸 1123 𝐸 1113 𝐸 1112 𝐸 1122 𝐸 2222 𝐸 2233 𝐸 2223 𝐸 1223 𝐸 1222 𝐸 1133 𝐸 2233 𝐸 3333 𝐸 2333 𝐸 1333 𝐸 1233 𝐸 1123 𝐸 2223 𝐸 2333 𝐸 2323 𝐸 2331 𝐸 2312 𝐸 1113 𝐸 1223 𝐸 1333 𝐸 2331 𝐸 1313 𝐸 3112 𝐸 1112 𝐸 1222 𝐸 1233 𝐸 2312 𝐸 3112 𝐸 1212

	𝐸 1111 𝐸 1122 𝐸 1133 √ 2 𝐸 1123 √ 2 𝐸 1113 √ 2 𝐸 1112	𝐸 1122 𝐸 2222 𝐸 2233 √ 2 𝐸 2223 √ 2 𝐸 1223 √ 2 𝐸 1222	𝐸 1133 𝐸 2233 𝐸 3333 √ 2 𝐸 2333 2𝐸 2323 √ 2 𝐸 1123 √ 2 𝐸 2223 √ 2 𝐸 2333 √ 2 𝐸 1333 2𝐸 2331 2 𝐸 1233 2𝐸 2312 √	√ 2 𝐸 1113 √ 2 𝐸 1223 √ 2 𝐸 1333 2𝐸 2331 2𝐸 1313 2𝐸 3112	√ √ √ 2𝐸 2312 2 𝐸 1112 2 𝐸 1222 2 𝐸 1233 2𝐸 3112 2𝐸 1212	⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎟ ⎟ ⎟	.	(4.3)
	matrix	⎛			⎞
	[E] =	⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝			⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎟ ⎟ ⎟	.			(4.4)

This matrix is called the Kelvin matrix and it is used, usually, for computations since it is the matrix representation of the elasticity tensor in an orthonormal basis (for instance, the matrix { E -1 } is simply equal to {E} -1 ). On the other side, if one simply aims at viewing the elasticity parameters, another notation can be used for the matrix of the elasticity tensor, called the Voigt

  𝑃 111 𝑃 122 𝑃 133 𝑃 123 𝑃 113 𝑃 112 𝑃 211 𝑃 222 𝑃 233 𝑃 223 𝑃 213 𝑃 212 𝑃 311 𝑃 322 𝑃 333 𝑃 323 𝑃 313 𝑃 312

		Ö						è
	[P] =									(4.9)
	or in so-called Kelvin notation (useful for computations since it is the matrix representation of
	the tensor in an orthonormal basis)
	{P} =	Ö 𝑃 111 𝑃 122 𝑃 133 𝑃 211 𝑃 222 𝑃 233 𝑃 311 𝑃 322 𝑃 333	√ √ √	2𝑃 123 2𝑃 223 2𝑃 323	√ √ √	2𝑃 113 2𝑃 213 2𝑃 313	√ √ √	2𝑃 112 2𝑃 212 2𝑃 312	è	.	(4.10)

  d𝜇; 𝜇 probability measure on R 𝑛 with support in 𝐾 } . Indeed, for 𝑥 𝑥 𝑥 ∈ 𝐾, if 𝛿 𝑥 𝑥 𝑥 denotes the probability Dirac measure on R 𝑛 at 𝑥 𝑥 𝑥, we have 𝑓 (𝑥 𝑥 𝑥) = ∫ 𝑓 d𝛿 𝑥 𝑥 𝑥 and, conversely, if 𝜇 is a probability measure on R 𝑛 with support in 𝐾, ∫ (called the moment matrix associated to 𝑦) and, if 𝑔 = ∑ 𝛽∈N 𝑛 𝑔 𝛽 𝑋 𝛽 , set

R 𝑛 𝑓 (𝑥 𝑥 𝑥)d𝜇(𝑥 𝑥 𝑥) ≥ ∫ R 𝑛 𝑓 * d𝜇(𝑥 𝑥 𝑥) = 𝑓 * . Now, if (𝑦 𝛼 ) 𝛼∈N 𝑛

is a sequence of real numbers, denote by 𝑀 (𝑦) the infinite symmetric matrix (𝑦 𝛼+𝛽 ) 𝛼,𝛽∈N 𝑛 ,

  Theorem 3.1 (Putinar, Jacobi-Prestel). Suppose that the polynomials 𝑔 1 , . . . , 𝑔 𝑚 describing 𝐾 satisfy the Archimedean hypothesis. Then, for all 𝑦 ∈ R N 𝑛 , 𝑦 has a representative measure on 𝐾 (i.e. there exists a finite Borel measure 𝜇 on R 𝑛 with support in 𝐾 such that, for any 𝛼 ∈ N 𝑛 , 𝑦 𝛼 = ∫ R 𝑛 𝑥 𝑥 𝑥 𝛼 d𝜇(𝑥 𝑥 𝑥)) if and only if the moment matrices 𝑀 (𝑦), 𝑀 (𝑔 1 •𝑦), . . . , 𝑀 (𝑔 𝑚 •𝑦) are positive semidefinite.

  𝑝𝑞𝑔) , which is represented by the localizing matrix 𝑀 (𝑔 • 𝑦) in the canonical basis of R[𝑋]. In particular, for every polynomial 𝑝, if p denotes the vector (𝑝 𝛼 ) 𝛼∈N 𝑛 , we have 𝑇 𝑀 (𝑔 • 𝑦) p ≥ 0 and, consequently, the linear mapping 𝐿 𝑦 has nonnegative values on M(𝑔 1 , . . . , 𝑔 𝑚 ). By Putinar's Positivstellensatz 3.3, this implies that 𝐿 𝑦 has nonnegative values on any polynomial 𝑝 ∈ R[𝑋] such that 𝑝(𝐾) ⊂]0; +∞[. If 𝑝 ∈ R[𝑋] satisfies 𝑝(𝐾) ⊂ [0; +∞[ then, for any positive real number 𝜖, the polynomial 𝑝 + 𝜖 has positive values on 𝐾 so that 𝐿 𝑦 (𝑝) + 𝜖 = 𝐿 𝑦 (𝑝 + 𝜖) ≥ 0, and therefore 𝐿 𝑦 (𝑝) ≥ 0. We can then apply Haviland's theorem ([31], see also [50, section 3.2] and Theorem 4.15 and section 4.6 of the up-to-date version of[44]) to the mapping 𝐿 𝑦 : there exists a measure 𝜇 on R 𝑛 with support in 𝐾 such that 𝐿 𝑦 (𝑝) = ∫ R

	𝐿 𝑦	(	𝑝 2 𝑔	)	= p

𝑛 𝑝(𝑥 𝑥 𝑥)d𝜇(𝑥 𝑥 𝑥) for all 𝑝 ∈ R[𝑋]. In particular, for all 𝛼 ∈ N 𝑛 , we have

  𝐸 1111 𝐸 1122 𝐸 1133 𝐸 1123 𝐸 1113 𝐸 1112 𝐸 1122 𝐸 2222 𝐸 2233 𝐸 2223 𝐸 1223 𝐸 1222 𝐸 1133 𝐸 2233 𝐸 3333 𝐸 2333 𝐸 1333 𝐸 1233 𝐸 1123 𝐸 2223 𝐸 2333 𝐸 2323 𝐸 2331 𝐸 2312 𝐸 1113 𝐸 1223 𝐸 1333 𝐸 2331 𝐸 1313 𝐸 3112 𝐸 1112 𝐸 1222 𝐸 1233 𝐸 2312 𝐸 3112 𝐸 1212

  36.401489, -20.227012, -38.908985, -6.396664, 27.780748, -2.277546, 44.251364, -4.557344, 21.161507).

  3 (see example 2.2), and set 𝐺 = O(3). A piezoelectricity tensor e ∈ Piez can be represented by a 3 × 6 matrix, in so-called Voigt representation, 122 e 133 e 123 e 113 e 112 e 211 e 222 e 233 e 223 e 213 e 212 e 311 e 322 e 333 e 323 e 313 e 312

	[e] =	⎛ ⎝ e 111 e

⎞

⎠ .

  [O -] ) 2 = min e∈Σ O - ‖e 0 -e‖ 2 , and e * ∈ Σ [O -] the closest to e 0 . In order to succeed, we first have to characterize the cubic piezoelectricity stratum Σ [O -] by polynomial equations (a proof of the following theorem is provided in Appendix C).Theorem 7.2. Let e = g + h ∈ Piez be a piezoelectricity tensor, with h ∈ H 3 (R 3 ) its leading harmonic part, let d 2 = h : h (i.e., (d 2 ) 𝑖𝑗 = ℎ 𝑖𝑘𝑙 ℎ 𝑘𝑙𝑗 ) , and d ′ 2 = d 2 -1 3 tr(d 2 ) q be second-order covariants of e. Then e ∈ Σ [O -] (is at least cubic) if and only if g = 0 and d ′ 2 = 0, and e ∈ Σ [O -] (is cubic) if and only if furthermore h ̸ = 0.

  3 ) has seven independent components and is represented by the following real matrix (in Voigt notation) 223ℎ 333 ℎ 223 ℎ 333 -𝐻 112ℎ 222 -ℎ 111ℎ 122 ℎ 123 111 , ℎ 112 , ℎ 122 , ℎ 123 , ℎ 222 , ℎ 223 , ℎ 333 ). , Σ [O -] ) 2 = min 𝑥 𝑥 𝑥∈ ̃ 𝐾 𝑓 (𝑥 𝑥 𝑥), where 𝑓 (𝑥 𝑥 𝑥) = ‖h 0 -h‖ 2 and

	(24) The traceless second order tensor d ′ [h] = ⎛ ⎝ ℎ 111 ℎ 122 -ℎ 111 -ℎ 122 ℎ 112 ℎ 222 -ℎ 112 -ℎ 222 -ℎ ⎞ ℎ 123 -ℎ 223 -ℎ 333 ℎ 112 ℎ 223 ℎ 123 ℎ 122 ⎠ 2 has five independent components (d ′ 2 ) 𝑖𝑗 detailed in Appen-dix B.
	We set
	𝑥 𝑥 𝑥 = (ℎ GloptiPoly computes the approximation
	(25) of the minimum ∆(e 0  𝐾 = For e 0 given by (22), we have (in C 2 /m 4 ) { 𝑥 𝑥 𝑥; d ′ 2 = 0, 𝑐 -𝑓 (𝑥 𝑥 𝑥) ≥ 0 ‖h 0 -h * ‖ 2	}	,	𝑐 = 3.
	𝑓 (𝑥 𝑥 𝑥) = 6ℎ 111 ℎ 122 + 6ℎ 223 ℎ 333 + 6ℎ 112 ℎ 222 + 4ℎ 2 111 + 6ℎ 2 112 + 6ℎ 2 122 + 4ℎ 2 222
	+ 6ℎ 2 223 + 4ℎ 2 333 + 6ℎ 2 123 -0.1002ℎ 111 -0.1742ℎ 122 + 0.1636ℎ 123 -0.0114ℎ 223 -5.2244ℎ 333 + 0.4574ℎ 112 + 0.0836ℎ 222 + 2.7367.

  2 , ‖e 0e * ‖ ‖e 0 ‖ = 0.684256.The results obtained for the raw piezoelectricity tensors e 𝑥 0 given in the Appendix D for wurzite Cr 𝑥 Al 1-𝑥 N, with different chromium concentrations, are summarized in Table1.

	𝑥 0 (AlN)	∆(E 0 , Σ [O -] ) 1.214681	‖e 0 -e * ‖ ‖e 0 ‖ 0.684256	Computation time (s) 0.7
	0.035	1.307327	0.715295	0.7
	0.07	1.364909	0.729065	0.8
	0.10	1.541726	0.785604	0.6
	0.13	1.542293	0.758240	1.0
	0.16	1.665883	0.793355	0.6
	0.19	1.852505	0.813719	0.7
	0.225	1.877377	0.781094	1.2
	0.255	1.944763	0.752770	0.7
	Table

  2 , 𝐼 6 := ‖𝑣 𝑣 𝑣 3 ‖ 2 , 𝐼 10 := ‖d ′ 2 × 𝑣 𝑣 𝑣 3 ‖ 2 , 𝐼 15 := det(𝑣 𝑣 𝑣 3 , 𝑣 𝑣 𝑣 5 , 𝑣 𝑣 𝑣 7 ).(2) A minimal integrity basis of R[H 3 (R 3 )] O(3) is constituted by the four invariants 𝐼 2 , 𝐼 4 , 𝐼 6 , and 𝐼 10 .Proof. To prove the theorem, it is enough to show that Smith and Bao's invariants can be expressed as polynomials of 𝐼 2 , 𝐼 4 , 𝐼 6 , 𝐼 10 , 𝐼 15 , since, then, this set will be generating and moreover the cardinal of a minimal integrity basis of homogeneous invariants does not depend on the choice of a particular basis[20]. Indeed, one can check that

	𝐾 4 = 𝐼 4 +	1 3	𝐼 2	2 ,
	𝐾 10 = -	4 3	𝐼 10 -	1 27	𝐼 2	3 𝐼 4 +	1 9	𝐼 2	2 𝐼 6 +	2 9	𝐼 2 𝐼 4	2 +	2 3	𝐼 4 𝐼 6 ,
	𝐾 15 = 2𝐼 15 ,
	which achieves the proof.														□
	Proof of theorem 7.2. Let e = g + h be a piezoelectricity tensor,
	g = (𝑣 𝑣 𝑣, 𝑤 𝑤 𝑤, a) ∈ H 1

  2 = 𝐼 2 (h), and thus an at least cubic tensor h 0 such that𝐼 2 (h 0 ) = 𝐼 2 (h), 𝐼 4 (h 0 ) = 𝐼 4 (h), 𝐼 6 (h 0 ) = 𝐼 6 (h), 𝐼 10 (h 0 ) = 𝐼 10 (h).But an integrity basis for a real representation of a compact group separate the orbits [2, Appendix C]. Hence, h and h 0 are necessarily in the same orbit, which means that h = 𝜌 3 (𝑔)h 0 , for some 𝑔 ∈ O(3). □

	0 [e 0.07 0 ] = ] = [e 0.10 0 ] = [e 0.13 0 ] = [e 0.16 0 ] = [e 0.19 0 ] = [e 0.225 0 ] = [e 0.255 0 ] =	⎛ ⎝ ⎛ ⎝ ⎛ ⎝ ⎛ ⎝ ⎛ ⎝ ⎛ ⎝ ⎛ ⎝ ⎛ ⎝	⎞ ⎠ , ⎞ -0.0329 0.0599 -0.0195 0.0267 -0.2327 -0.0988 -0.0548 -0.0129 -0.0063 -0.2075 -0.0051 -0.0293 -0.5872 -0.4900 1.5560 -0.0218 -0.0278 -0.0115 -0.0393 0.0185 0.0048 0.0290 -0.2171 -0.0436 -0.07 0.0044 ⎠ , 0.0554 0.0137 -0.1574 0.0198 -0.5179 -0.5886 1.6521 -0.0085 -0.0095 -0.0119 ⎞ 0.0291 -0.0141 -0.0523 -0.0016 0.0028 0.0138 ⎠ , -0.0611 0.0819 -0.0567 -0.1841 0.0116 0.0270 -0.5244 -0.5918 1.7715 -0.0018 0.0066 -0.0145 -0.0985 0.1138 0.047 ⎞ -0.0169 -0.0169 -0.0984 0.0558 ⎠ , 0.0183 -0.0367 -0.1735 -0.0384 0.0474 -0.5441 -0.5455 1.8506 -0.0148 -0.0016 -0.0193 ⎞ 0.0315 -0.0375 0.0273 0.0206 0.0206 0.0933 ⎠ , -0.215 -0.0717 0.0845 -0.2157 0.0438 -0.0332 -0.4517 -0.5587 1.9243 0.0447 0.0277 -0.0482 0.4524 ⎞ 0.3564 -0.0827 -0.0276 -0.0276 0.1067 -0.0783 0.0868 0.0318 ⎠ , 0.0037 -0.1053 -0.0765 -0.5768 -0.4566 2.0350 -0.1332 -0.1016 -0.1253 0.0428 ⎞ 0.0974 -0.0429 -0.0319 -0.0363 0.0099 ⎠ , -0.1399 -0.2386 -0.0253 -0.1505 0.0143 -0.1770 -0.5800 -0.5552 2.2197 0.0164 0.0048 0.0234 ⎞ -0.0914 0.0758 0.0000 -0.0022 -0.2835 0.0000 -0.6063 -0.5847 2.3709 -0.0714 -0.0738 -0.0559 0.0000 -0.0022 0.0000 -0.2660 0.0002 -0.0020 ⎠ .

  𝐸 1111 𝐸 1122 𝐸 1133 𝐸 1123 𝐸 1113 𝐸 1112 𝐸 1122 𝐸 2222 𝐸 2233 𝐸 2223 𝐸 1223 𝐸 1222 𝐸 1133 𝐸 2233 𝐸 3333 𝐸 2333 𝐸 1333 𝐸 1233 𝐸 1123 𝐸 2223 𝐸 2333 𝐸 2323 𝐸 2331 𝐸 2312 𝐸 1113 𝐸 1223 𝐸 1333 𝐸 2331 𝐸 1313 𝐸 3112 𝐸 1112 𝐸 1222 𝐸 1233 𝐸 2312 𝐸 3112 𝐸 1212If E has at least the cubic symmetry, there exists an orthonormal basis (𝑒 𝑒 𝑒 𝑖 ) (called the natural basis or the cubic basis), in which E has the so-called cubic normal form in Voigt representation(𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 ).If 𝐸 1111 -𝐸 1122 -2𝐸 1212 = 0, then E is isotropic. Otherwise, it is cubic. One may point out the cubic symmetry group (O) and write E O for the normal form of a cubic tensor E.If 𝐸, 𝜈 and 𝐺, respectively denote the Young modulus, the Poisson ratio, and the shear modulus of a material, the engineer's expressions for the 𝐸 𝑖𝑗𝑘𝑙 are

	(8)	[E] =	⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝	𝐸 1111 𝐸 1122 𝐸 1122 𝐸 1122 𝐸 1111 𝐸 1122 𝐸 1122 𝐸 1122 𝐸 1111 0 0 0 0 0 0	0 0 0 𝐸 1212 0	0 0 0 0 𝐸 1212	0 0 0 0 0	⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
				0	0	0	0	0	𝐸 1212
									⎞
									⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎟	.

  1 2 (𝛿 𝑖𝑘 𝛿 𝑗𝑙 + 𝛿 𝑖𝑙 𝛿 𝑗𝑘 ) and H ∈ H 4 is a cubic fourth-order harmonic tensor 2 . Here,

		𝜆 =	1 15	(2 tr(tr 12 E) -tr(tr 13 E)) =	1 5	(𝐸 1111 -2𝐸 1212 + 4𝐸 1122 ),
	and	𝜇 =	1 30	(-tr(tr 12 E) + 3 tr(tr 13 E)) =	1 5	(𝐸 1111 + 3𝐸 1212 -𝐸 1122 ),
	are the Lamé constants.

  Remark 3.4. The decomposition of any elasticity tensor provided by the above formulas,E = (𝜆, 𝜇, d ′ , v ′ , H),into the harmonic components 𝜆, 𝜇 ∈ H 0 , d ′ , v ′ ∈ H 2 and H ∈ H 4 , is the so-called harmonic decomposition of E[10, 58, 18].

′ ) and d 2 = H . . . H, with components (d 2 ) 𝑖𝑗 = 𝐻 𝑖𝑝𝑞𝑟 𝐻 𝑝𝑞𝑟𝑗 . Then, E is cubic if and only if d ′ = v ′ = 0, d ′ 2 = 0, and 𝐽 2 = tr d 2 ̸ = 0.

  𝐹 (H, 𝜆 𝜆 𝜆) = ‖H 0 -H‖ 2 + 𝜆 𝜆 𝜆 : 𝑔(H), with H ∈ H 4 an harmonic fourth-order tensor and where the Lagrange multiplier 𝜆 𝜆 𝜆 ∈ H 𝑠′ ∈ H 4 , is the fourth-order harmonic part of the tensor (H • 𝜆 𝜆 𝜆) 𝑠 (of components 𝐻 𝑖𝑗𝑘𝑝 𝜆 𝑝𝑙 ). It can be computed using for example Eq. (17), or using directly the harmonic decomposition of totally symmetric tensors[58][52, Section 2.2], with here 𝜆 𝜆 𝜆 and tr tr (S(𝜆 𝜆 𝜆)) = tr tr(H • 𝜆 𝜆 𝜆) 𝑠 = 0, 𝐻 𝑖𝑗𝑘𝑙 of H ∈ H 4 and the 5 independent components 𝜆 𝑖𝑗 of 𝜆 𝜆 𝜆 ∈ H 2 .The system (23) can be further simplified by extracting from the equality H -H 0 + S(𝜆 𝜆 𝜆) = 0 some linear equations in H. 𝐻 𝑖𝑝𝑞𝑟 𝐻 𝑗𝑝𝑞𝑠 𝜆 𝑟𝑠 , (H 2 : 𝜆 𝜆 𝜆) 𝑖𝑗 = 𝐻 𝑖𝑗𝑘𝑙 𝐻 𝑘𝑙𝑚𝑛 𝜆 𝑚𝑛 .If H is at least cubic, then, d ′ 2 = 0 and 𝜆 𝜆 𝜆 • d 2 = d 2 • 𝜆 𝜆 𝜆 is symmetric. Thus

	(21) H : so that (introducing the symmetrized tensor product ⊙) tr(H • 𝜆 𝜆 𝜆) 𝑠 = 1 2 S(𝜆 𝜆 𝜆) = (H • 𝜆 𝜆 𝜆) 𝑠 -3 7 1 ⊙ (H : 𝜆 𝜆 𝜆). (22)
	Therefore, the Euler-Lagrange equations equations {	𝜕𝐹 𝜕H	= 0 and	𝜕𝐹 𝜕𝜆 𝜆 𝜆	= 0 reduce to the system of
	(23)				H -H 0 + S(𝜆 𝜆 𝜆) = 0 (9 scalar equations), d ′ 2 = 0 (5 scalar equations),
	in the 9 independent components Lemma 5.1. The Euler-Lagrange system (23) implies that (24) ⎧ ⎪ ⎨ ⎪ ⎩ H . . . H = 0, (3 linear scalar equations) . . . H 0 -H 0 (1 quadratic scalar equation) (H -H 0 ) :: H = 0, d ′ 2 = 0. (5 quadratic scalar equations)
	(25a)											. . . H = 0,
	(25b)				H	. . . H -H	. . . H 0 + H	. . . S(𝜆 𝜆 𝜆) = 0.
	By (22) and some calculations, we have
		S(𝜆 𝜆 𝜆)	. . . H = (H • 𝜆 𝜆 𝜆) 𝑠 . . . H -	3 7	(1 ⊙ (H : 𝜆 𝜆 𝜆))	. . . H =	1 4	𝜆 𝜆 𝜆 • d 2 +	3 4	c -	3 14	H 2 : 𝜆 𝜆 𝜆
	and	H	. . . S(𝜆 𝜆 𝜆) = H	. . . (H • 𝜆 𝜆 𝜆) 𝑠 -	3 7 H S(𝜆 𝜆 𝜆) H . . . (1 ⊙ (H : 𝜆 𝜆 𝜆)) = 1 4 d 2 • 𝜆 𝜆 𝜆 + 3 4 c -3 14 . . . H = H . . . S(𝜆 𝜆 𝜆),
	and, substracting (25b) from (25a), we get harmonic (deviatoric) second-order tensor. The differential of 𝐹 with respect to H is given by H . . . H = 0. . . . H 0 -H 0 The second equation in (24) is obtained, by applying the Euler lemma on homogeneous 2 is an . . . 𝛿H + 𝛿H . . . H) functions to the quadratic function d𝐹.𝛿H = 2(H -H 0 ) :: 𝛿H + 𝜆 𝜆 𝜆 : (H H ↦ → 1 2 𝜆 𝜆 𝜆 : d ′ 2 (H),

= 2(H -H 0 ) :: 𝛿H + 2 S(𝜆 𝜆 𝜆) :: 𝛿H, thanks to the equalities 𝜆 𝜆 𝜆 : d ′ 2 = 𝜆 𝜆 𝜆 : d 2 = 𝜆 𝜆 𝜆 : (H . . . H), where (20) S(𝜆 𝜆 𝜆) := 1 2 grad H (𝜆 𝜆 𝜆 : d ′ 2 ) = (H • 𝜆 𝜆 𝜆) Proof. By contracting three times the first equation H -H 0 + S(𝜆 𝜆 𝜆) = 0 in (23) with H on the right and then on the left, we get H . . . H -H 0 . . . H + S(𝜆 𝜆 𝜆) 2 : 𝜆 𝜆 𝜆 where both c and H 2 : 𝜆 𝜆 𝜆 = H : H : 𝜆 𝜆 𝜆 are symmetric second-order tensors with components 𝑐 𝑖𝑗 = whose gradient is S(𝜆 𝜆 𝜆). We get S(𝜆 𝜆 𝜆) :: H = 1 2 ( grad H (𝜆 𝜆 𝜆 : d ′ 2 )

3 ,

 3 𝑋 1 = -6.396655, 𝑋 2 = 27.780761, 𝑌 1 = -2.277535, 𝑌 2 = 44.251233, 𝑍 1 = -4.557361, 𝑍 2 = 21.161420, Λ 1 = -36.401302, Λ 2 = -20.226895, Λ 3 = -38.908854, for H * . The numerical value 𝑓 (0) = ‖H 0 ‖ 2 = 57886.9 GPa 2 for H isotropic is found larger than the one 2530.47 GPa 2 for the optimal cubic tensor H. With the values 𝜆 = 𝜆 0 = 1583/15 = 105.533333 and 𝜇 = 𝜇 0 = 1453/15 = 96.866667, the tensor E * = 2𝜇 0 I + 𝜆 0 1 ⊗ 1 + H * ,

	of Voigt representation (28) [E * ] = ⎛ ⎜ ⎜ ⎜ ⎝ 240.130916 144.442188 125.760229 6.39665526 41.9736976 -21.1614201	144.442188 223.956510 141.934636 -27.7807617 2.27753546 16.6040582	125.760229 141.934636 242.638469 21.3841064 4.55736193 -44.2512331	6.39665526 -27.7807617 21.3841064 133.267969 2.27753546 4.55736193	41.9736976 2.27753546 -44.2512331 4.55736193 6.39665526 117.093562	-21.1614201 16.6040582 4.55736193 2.27753546 135.775521 6.39665526	⎞ ⎟ ⎟ ⎟ ⎠	GPa
	is the (cubic) elasticity tensor that minimizes the distance to cubic symmetry, with then
		𝑑(E 0 , cubic symmetry) = 74.13 GPa.
	With a relative distance	‖E 0

  to(15),𝐽 2 = ‖H * ‖ 2 = 55356.440 GPa 2 , 𝐽 3 = tr 13 (H * 3 ) = -2377889.1 GPa 3 , (𝑒 𝑒 𝑒 1 ,𝑒 𝑒 𝑒 2 ,𝑒 𝑒 𝑒 3 )

	so that, in GPa,
			𝜆 = 105.533333,	𝜇 = 96.866667,	𝛿 =	𝐽 3 4𝐽 2	= -10.738990,
	and
	(29)	⎛	213.354743 148.489295 148.489295	0		0	0	⎞
	[E * O ] =	⎜ ⎜ ⎜ ⎜ ⎝	148.489295 213.354743 148.489295 148.489295 148.489295 213.354743 0 0 0 0 0 0	0 0 139.822628 0	0 0 0 139.822628	0 0 0 0	⎟ ⎟ ⎟ ⎟ ⎠
			0	0	0	0		0	139.822628

  𝜎 22 ) 2 + (𝜎 33 -𝜎 11 ) 2 + (𝜎 22 -𝜎 33 ) 2 )

				, setting
	(31)	𝐶 :=	2 3	(𝐹 1111 -𝐹 1122 ) and 𝐿 :=	3𝐹 1212 𝐹 1111 -𝐹 1122	,
					+ 2𝐿	(	𝜎 2 12 + 𝜎 2 13 + 𝜎 2 23

allows to recover the standard expression of cubic Hill yield criterion (in cubic basis (𝑒 𝑒 𝑒 𝑖 )), as

𝜎 𝜎 𝜎 ′ : P : 𝜎 𝜎 𝜎 ′ = 1 2

( (𝜎 11 -

  * H ** ,

	the Voigt representation
	[F ** ] =	⎛ ⎜ ⎜ ⎜ ⎝	211.420595 -106.652774 -52.101155 -35.826922 -72.960965 64.693254	64.693254 -18.805804 -47.517810 -106.652774 -52.101155 -35.826922 -72.960965 257.771914 -98.452474 83.635836 -98.452474 203.220295 -47.808914 91.766769 -17.175444 83.635836 -47.808914 89.880860 -17.175444 -18.805804 -18.805804 91.766769 -47.517810 -17.175444 -18.805804 -35.826922 81.680560 -17.175444 136.232178 -35.826922	⎞ ⎟ ⎟ ⎟ ⎠	.

  2 2 × d 2 (a third order tensor defined by (3)) are given in [2, Appendix B]. Note that × d 2 ‖ 2 (3 I -1 ⊗ 1) . Remark B.2. The fourth order tensor C 0 is a rational covariant of H 0 , therefore of E 0 . It is built from the second-order quadratic covariant d 2 = d 2 (E 0 ). When d 2 is orthotropic (which is generically the case for the raw/measured tensor E 0 ) the third-order tensor d 2 2 ×d 2 is tetrahedral and harmonic, and the fourth-order tensor (d 2 2 × d 2 ) • (d 2 2 × d 2

						‖d 2 2 × d 2 ‖ 2 =	1 12	( (	tr(d ′ 2 2 ) ) 3 -6	(	tr(d ′ 3 2 )	) 2	)	.
	and	[( d 2 2 × d 2	)	•	(	d 2 2 × d 2	)] ′ =	(	d 2 2 × d 2	)	•	(	d 2 2 × d 2	)	-	1 15	‖d 2 2

Table 1 .

 1 0 , cubic symmetry) 𝑀 (E 0 , [O]) ∆(E 0 , [O]) Comparison of upper bounds estimates of the distance to cubic elasticity.[𝐻] [1] [Z 2 ] [Z 3 ] [Z 4 ] [D 2 ] [D 3 ] [D 4 ] [SO(2)] [O(2)] [O(3)]Table A.1: Dimension of the fixed-point sets for each symmetry class of 𝒫iez of type I Table A.2: Dimension of the fixed-point sets for each symmetry class of 𝒫iez of type II

	Distance/estimates (GPa)	74.13	245.5	238.6
	Relative distance/estimates	0.1039	0.3441	0.3344

Table A .

 A 3: Dimension of the fixed-point sets for each symmetry class of 𝒫iez of type III Let (E, P, S) ∈ 𝒫iez, E, P and S can be represented by the following matrices (in Voigt 𝐸 1111 𝐸 1122 𝐸 1133 𝐸 1123 𝐸 1131 𝐸 1112 𝐸 1122 𝐸 2222 𝐸 2233 𝐸 2223 𝐸 2231 𝐸 2212 𝐸 1133 𝐸 2233 𝐸 3333 𝐸 3323 𝐸 3331 𝐸 3312 𝐸 1123 𝐸 2223 𝐸 3323 𝐸 2323 𝐸 2331 𝐸 2312 𝐸 1131 𝐸 2231 𝐸 3331 𝐸 2331 𝐸 1313 𝐸 3112 𝐸 1112 𝐸 2212 𝐸 3312 𝐸 2312 𝐸 3112 𝐸 1212 Ö 𝑃 111 𝑃 122 𝑃 133 𝑃 123 𝑃 113 𝑃 112 𝑃 211 𝑃 222 𝑃 233 𝑃 223 𝑃 213 𝑃 212 𝑃 311 𝑃 322 𝑃 333 𝑃 323 𝑃 313 𝑃 312 𝑆 11 𝑆 12 𝑆 13 𝑆 12 𝑆 22 𝑆 23 𝑆 13 𝑆 23 𝑆 33

	notation):	⎛	⎞
	[E] =	⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝	⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎟ ⎟ ⎟	,
	[P] =

è and [S] = Ö

A set is constructible if it is Zariski open in its Zariski closure.

P. AZZI, R. DESMORAT, J. GRIVAUX, AND B. KOLEV

Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 place Jussieu, 75005, Paris, France Email address: perla.azzi@ens-paris-saclay.fr (Rodrigue Desmorat) Université Paris-Saclay, ENS Paris-Saclay, CentraleSupélec, CNRS, LMPS -Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, FranceEmail address: rodrigue.desmorat@ens-paris-saclay.fr (Julien Grivaux) Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 place Jussieu, 75005, Paris, France Email address: julien.grivaux@imj-prg.fr (Boris Kolev) Université Paris-Saclay, ENS Paris-Saclay, CentraleSupélec, CNRS, LMPS -Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, FranceEmail address: boris.kolev@ens-paris-saclay.fr

A tensor of order 𝑛 is a 𝑛-linear map between vector spaces, denoted by 𝑛 indices. A tensor of order 𝑛 is totally symmetric if it is invariant under permutation of its 𝑛 indices.

relating induced polarization in a dielectric material to the strain tensor.

(Boris Kolev) Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.Email address: boris.kolev@ens-paris-saclay.fr (Fabien Priziac) Univ Bretagne Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France Email address: fabien.priziac@univ-ubs.fr

A symmetry stratum is the set of all tensors which have the same symmetry class.

i.e. totally symmetric, H = H 𝑠 , and traceless, tr𝑖𝑗 H = 0.

Remerciements

Mes premiers remerciements s'adressent à mon équipe d'encadrement Julien Grivaux, Rodrigue

The authors were partially supported by CNRS Projet 80-Prime GAMM (Géométrie algébrique complexe/réelle et mécanique des matériaux).

Theorem 3.2.7. Let H ∈ H 4 (R 3 ) a fourth order harmonic tensor. Then

(1) H ∈ Σ [SO (3)] ⇐⇒ d ′ 2 = 0.

( Proof.

1. For the proof of (1 ) and (2 ) see [91, Hence, H ∈ Σ [O (2)] in all cases. = 0 implies that d 2 is either isotropic or transversely isotropic [91, lemma 8.1]:

If

if 𝑚 even and 4 ∤ 𝑚

else

Table 3.3: Clips between type II and III O(3)-subgroups (part 2)

3.5

Clips operations between type II and type III O(3)-subgroups:

) be a subgroup of type III (see section 3.1) and 𝐻 a subgroup of SO (3). By definition, the clips operation between a type II and a type III subgroups

Indeed, as we have

In the following, we prove a theorem which describes the possible conjugacy classes belonging to the clips between a type II and type III subgroups of O(3). The computation of such classes can be complicated since it involves intersection of groups of type III which can be tricky sometimes

Lemma 3.5.15. For any integer 𝑛 ≥ 2 and 𝑑 𝑘 = gcd(𝑛, 𝑘) for 𝑘 = 2, 3, we have

} .

Proof. We deduce from [86, table 1] and theorem 3.5.2 that the classes in

, corresponding to type I subgroups, are in the following list

and the classes corresponding to type III subgroups are in the set

} .

We can check that all the eventualities can occur by using (3.18):

where D 𝑛 and 𝛾D 𝑛 are given in (3.10) and (3.22) and T in (3.11). We get

and the classes corresponding to type III subgroups are in the set

} .

We can check that all the eventualities can occur except [D 𝑧 3 ] (same argument as in lemma 3.5.11) by using (3.18):

where 𝛾T is given by (3.25) [T] for the identity rotation for instance.

Lemma 3.5.24. We have

} .

Proof. By theorem 3.5.2 we deduce that the classes in the clips

} .

All the classes of the above list can be realized by a rotation 𝑔 using the union

. Indeed, we get:

for the identity rotation for instance.

Lemma 3.5.25. We have

} .

The symmetry classes for O(3)-representation on Ela and S are the same as the symmetry classes for SO(3)-representation which can be found in [38] and [START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF], except that each type I subgroup occurring in the list of isotropy classes has to be replaced by the corresponding type II subgroup (see section 3.1). Indeed, -I acts trivially on Ela and S. Hence, the isotropy classes for the O(3)-representation on Ela and S are given by

We deduce the isotropy classes of the Piezoelectricity coupled law 𝒫iez = Ela ⊕ Piez ⊕ S from lemma 3.3.2 by calculating the clips operations between the isotropy classes of Ela, Piez and S (see Table 3.2 and Table 3.3 for clips between type II and III O(3)-subgroups and [86, table 1] for clips between two type I and remark 3.1 (1) for type I with type II and two type II).

DISTANCE TO A CONSTITUTIVE TENSOR ISOTROPY STRATUM BY LASSERRE POLYNOMIAL OPTIMIZATION METHOD

P. AZZI, R. DESMORAT, B. KOLEV, AND F. PRIZIAC Abstract. We give a detailed description of a polynomial optimization method allowing to solve a problem in continuum mechanics: the determination of the elasticity or the piezoelectricity tensor of a specific isotropy stratum the closest to a given experimental tensor, and the calculation of the distance to the given tensor from the considered isotropy stratum. We take advantage of the fact that the isotropy strata are semialgebraic sets to show that the method, developed by Lasserre and coworkers which consists in solving polynomial optimization problems with semialgebraic constraints, successfully applies.

H the fourth-order harmonic tensor given by ( 18), are: Smith and Bao [67] have derived a minimal integrity basis of five invariants for the algebra R[H 3 (R 3 )] SO (3) , of polynomial SO(3)-invariants of the third-order harmonic tensors h ∈ H 3 (R 3 ). These five invariants (equations (2.3) and (2.4) in [67]) can be recast in a more intrinsic form as

where d 2 = h : h and 𝑣 𝑣 𝑣 3 := h : d ′ 2 . In [59], Olive and Auffray have used these results to deduce that a minimal integrity basis for the algebra R[H 3 (R 3 )] O (3) , of polynomial O(3)-invariants of h ∈ H 3 (R 3 ) consists of the four invariants 𝐼 2 , 𝐾 4 , 𝐼 6 , and 𝐾 10 .

Appendix D. Raw piezoelectricity tensors for wurtzite

The raw piezoelectricity tensors e 𝑥 0 considered in section 7 correspond to the mean values computed in [49] for wurtzite Cr 𝑥 Al 1-𝑥 N, with 𝑥 the chromium concentration (in C/m 2 ), [e 0.035 P. AZZI, R. DESMORAT, B. KOLEV, AND F. PRIZIAC Abstract. Generically, a fully measured elasticity tensor has no material symmetry. For single crystals with a cubic lattice, or for the aeronautics turbine blades superalloys such as Nickelbased CMSX-4, cubic symmetry is nevertheless expected. It is in practice necessary to compute the nearest cubic elasticity tensor to a given raw one. Mathematically formulated, the problem consists in finding the distance between a given tensor and the cubic symmetry stratum.

It is known that closed symmetry strata (for any tensorial representation of the rotation group) are semialgebraic sets, defined by polynomial equations and inequalities. It has been recently shown that the closed cubic elasticity stratum is moreover algebraic, which means that it can be defined by polynomial equations only (without requirement to polynomial inequalities). We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution using the technique of Gröbner bases. The proposed methodology also applies to cubic Hill elasto-plasticity (where two fourth-order constitutive tensors are involved).

Introduction

Anisotropic elasto-plasticity theories introduce (at least) two fourth-order constitutive tensors, the Hooke and the Hill tensors for instance. It is nowadays possible to measure/determine all their components [41,8,7,27,29,16,29,46]. These measured constitutive tensors are however generically triclinic (they have no material symmetry).

On the other hand, many materials (such as composite/engineered materials, single crystal superalloys or rocks) have an expected symmetry, most often due to their microstructure and their elaboration process. In practice, appealing to Curie principle ("the symmetries of the causes are to be found in the effects"), their constitutive tensors shall inherit the material symmetry (orthotropy, cubic or monoclinic symmetry for example), so that the natural question is to determine the constitutive tensor with a given material symmetry the nearest to a given measured (triclinic) constitutive tensor. This question has been extensively studied, from both the theoretical and numerical points of view, since the pioneering work of Gazis, Tadjbakhsh and Toupin [31], and subsequent works in the 90s [8,7,27,22,29]. Most works focus on the elasticity tensor [31,25,27,28,34,22,29,21,47,40,24,48], a few ones on the piezoelectricity tensor [66]. So far, we are not aware of some similar studies for the Hill plasticity tensor or the combination of several constitutive tensors.

Even if some analytical attempts exist [31,63,59,6], the distance to an elasticity symmetry class problem is usually solved numerically, following [27,29,22,21], using the parameterization of a symmetry class, by its normal form A (for instance (8) for cubic symmetry [25]) and a rotation 𝑄, E = 𝑄 ⋆ A with components 𝐸 𝑖𝑗𝑘𝑙 = 𝑄 𝑖𝑝 𝑄 𝑗𝑞 𝑄 𝑘𝑟 𝑄 𝑙𝑠 𝐴 𝑝𝑞𝑟𝑠 where ⋆ stands for the action of the rotation 𝑄 on the tensor A [26,63]. Letting E 0 be the given experimental (raw) elasticity tensor, one has thus to solve the minimization problem (1) min

c) and of the 5 equations 𝑔 𝑖𝑗 = (d 2 ) ′ 𝑖𝑗 = 0, with

Using the first three linear equations (of point (a)), we further reduce the system to 6 equations 𝑔 𝑖𝑗 = 0 and 𝑔 6 = (H -H 0 ) :: H = 0, quadratic in the 6 variables 𝑌 2 , 𝑍 1 , 𝑍 2 , Λ 1 , Λ 2 , Λ 3 , and which can be solved thanks to the determination of a Gröbner basis GB, by symbolic computation using Mathematica software 3 . We take advantage of the fact that the material parameters (here the components of E 0 ), are measured with only a few significant digits to work with rational coefficients polynomials. This point is of main importance in the resolution of a system of polynomial equations by the obtention of a Gröbner basis (see remark A.1 of the Appendix). The result is a set GB = {GB 1 , . . . , GB 32 } of 32 polynomials GB 𝑛 (unfortunately too lengthy to be given) in the variables 𝑌 2 , 𝑍 1 , 𝑍 2 , Λ 1 , Λ 2 , Λ 3 , and which vanishes if and only if the initial (polynomial) system ( 24) is satisfied.

In the present application, the first polynomial of the Gröbner basis GB 1 is found to be function of Λ 3 only, GB 2 function of Λ 2 and Λ 3 (but linear in Λ 2 ), and so on, up to GB 32 function of all the variables (but linear in 𝑌 2 ), as in (38) of Appendix A with 𝑛 = 6 and 𝑥 6 = Λ 3 . Solving 4 GB 1 (Λ 3 ) = 0, we get either Λ 3 = Λ (0) 3 = 0 (leading to the isotropic solution H (0) = 0) or Λ 3 is a real root of a polynomial of degree 14, which has 8 non-zero real roots (in practice determined with a 50 significant digits precision), Λ

3 = -10.425971, Λ

3 = -6.225368, Λ

3 = -3.056232, Λ

3 = 1.745698, Λ

3 = 13.541284. Except from this initial (roots) solving, the remaining unknowns Λ 2 , then Λ 1 , 𝑍 2 , 𝑍 1 and last 𝑌 2 , are obtained analytically one per one for each Λ (𝑠) 3 solution (thanks to the equations GB 𝑚 = 0, 𝑚 ≥ 2, given by the elements of the Gröbner basis GB, when Λ 3 is evaluated). The variables 𝑋 1 , 𝑋 2 , 𝑌 1 are finally given by the three linear equations of point (a).

This polynomial optimization approach shows that, generically, for the distance to cubic symmetry problem, the number of critical points solutions of the first-order Euler-Lagrange equations ( 24) is finite, the corresponding solutions H (𝑠) being fully determined by all the roots of the polynomials in the Gröbner basis GB. The global minimum min 𝑓 (H) is simply the minimum minimorum min

3 by the command GB = GroebnerBasis [{𝑔11, 𝑔22, 𝑔12, 𝑔13, 𝑔23, 𝑔6} , {𝑌 2, 𝑍1, 𝑍2, Λ1, Λ2, Λ3}], where by default the lexicographic elimination order is used. 4 using the command NSolve[GB[ [1]] == 0, Λ3, WorkingPrecision → 50].

Appendix A. Solving algebraic systems using Gröbner bases

In this appendix, we propose to explain how to use Gröbner bases to solve non-linear algebraic systems. Our goal is not to summarize the theory of Gröbner bases, nor to introduce the basics of algebraic geometry but to explain through some examples how it works. For more details on this topic and a deeper insight, we strongly recommend the following books [19,60], which contain a lot of references.

Gröbner bases were introduced in the sixties by Buchberger [17]. Like Gaussian elimination method is used to solve a system of linear equations, Gröbner bases are useful to solve a system of non-linear algebraic equations ( 36)

where 𝑓 1 , . . . , 𝑓 𝑚 are polynomial functions in the variables 𝑥 1 , . . . , 𝑥 𝑛 . Note however that in general, and even for one variable, it is useless to search for closed-form solutions. Therefore, what is expected is a procedure which produces a new system of algebraic equations which is simpler. Contrary to Gauss elimination algorithm, where the variables are naturally ordered by the choice of a basis, we need to choose a total order on monomials in order to make the Gröbner bases algorithm to work. There are many total orders on monomials in several variables, the most common being the lexicographic order induced by 𝑥 1 < 𝑥 2 < • • • < 𝑥 𝑛 , and the resulting Gröbner basis will depend drastically on the choice of an order.

Let us illustrate what we mean here through an example. Consider, for instance, an intersection of three quadrics in R 3 , given by the following non-linear system of three homogeneous polynomial equations of degree 2 (37)

in the three variables (𝑥 1 , 𝑥 2 , 𝑥 3 ). The computation of a Gröbner basis for this system (with the lexicographic order induced by 𝑥 1 < 𝑥 2 < 𝑥 3 ) leads to the following equivalent system of equations

Note

Then, a Gröbner basis (computed using the lexicographic order induced by 𝑥 1 < 𝑥 2 < • • • < 𝑥 𝑛 ) provides a new system of generators of 𝐼 which is compatible with the sequence of elimination ideals. Let us illustrate what we mean here, using our first example (37). In that case, the following Gröbner basis was computed

In this example, 𝐼 is generated by 𝑔 1 , 𝑔 2 , 𝑔 3 , 𝐼 1 by 𝑔 2 , 𝑔 3 and 𝐼 2 by 𝑔 3 . It is in this sense that a Gröbner basis can be considered as a triangulation of the initial problem. Hence, in this example, solving the problem consists first in finding the roots of 𝑔 3 = 0, then calculating 𝑥 2 using 𝑔 2 = 0 and then 𝑥 1 using 𝑔 1 = 0. Consider now the second example (39). In that case, the following Gröbner basis was computed using the lexicographic order

In this example, 𝐼 is generated by 𝑔 1 , 𝑔 2 , 𝑔 3 , 𝑔 4 , 𝐼 1 by 𝑔 4 and 𝐼 2 by 0. We could continue here to explain the complete resolution of the problem but it appears that changing the order on monomials makes the resolution by far much more readable for a human being. Indeed, changing the lexicographic order 𝑥 1 < 𝑥 2 < 𝑥 3 to 𝑥 3 < 𝑥 1 < 𝑥 2 leads to the following Gröbner basis

, and, then, 𝐼 is generated by 𝑔 1 , 𝑔 2 , 𝐼 1 := 𝐼 ∩ C[𝑥 1 , 𝑥 2 ] by 𝑔 2 and 𝐼 2 := 𝐼 ∩ C[𝑥 2 ] by 0. We will now proceed to the complete resolution of the system. First, we need to solve the equation in one variable 𝑥 2 given by 𝐼 2 . Since 𝐼 2 is generated by 0, this means that the variable 𝑥 2 is free. We will thus set 𝑥 2 = 𝑡 (𝑡 ∈ C). Then, we need to solve the system of equations in two variables (𝑥 2 , 𝑥 1 ) given by 𝐼 1 . More precisely, since we have already solved the problem for 𝑥 2 (the system is triangular), we seek solutions (𝑥 2 , 𝑥 1 ) of 𝑥 2 1 -𝑥 1 𝑥 2 + 𝑥 2 2 = 0, which extend the solution 𝑥 2 = 𝑡. Hence, 𝑥 2 is no more a variable here but a parameter. This equation has either one solution (𝑥 2 = 0, 𝑥 1 = 0) if 𝑡 = 0 or two conjugate imaginary solutions if 𝑡 ̸ = 0. Now, we need to solve the system of equations in three variables (𝑥 2 , 𝑥 1 , 𝑥 3 ) given by 𝐼. Hence, we need to solve the equation 𝑥 2 3 -1 + 𝑥 1 𝑥 2 = 0 but where (𝑥 2 = 𝑎, 𝑥 1 = 𝑏) is a solution of the previous step and where 𝑥 2 = 𝑎 and 𝑥 1 = 𝑏 should be considered as parameters of the problem. Such a solution is said to extend the previous one. In our example, we find exactly two solutions for 𝑥 3 for each solution (𝑥 2 = 𝑎, 𝑥 1 = 𝑏) (because the coefficient of 𝑥 2 3 is one). In other examples, nevertheless, some solutions could be not extendable (for example, if the coefficient of 𝑥 2 3 depends on 𝑎, 𝑏 and vanishes for some values of 𝑎 and 𝑏). This example illustrate the triangular process allowed by the computation of a Gröbner basis in solving non-linear algebraic equations. Note finally that solutions are sought in C. In this example, there are an infinite number of complex solutions. Of course, it may happen that there are no real solution at all. Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 place Jussieu, 75005, Paris, France Email address, Perla Azzi: perla.azzi@ens-paris-saclay.fr (Rodrigue Desmorat) Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.

Email address: rodrigue.desmorat@ens-paris-saclay.fr (Boris Kolev) Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France.

Email address: boris.kolev@math.cnrs.fr (Fabien Priziac) Univ Bretagne Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France Email address: fabien.priziac@univ-ubs.fr