Boris Desmorat

Marc Olive

Nicolas Auffray

Rodrigue Desmorat

Boris Kolev

Kolev

Computation of minimal covariants bases for 2D coupled constitutive laws

Keywords: Mathematics Subject Classification. 74B05, 15A72, 74-04 Constitutive tensors, Tensor invariants, Integrity basis, Higher-order tensors, Polynomial covariants, Harmonic decomposition. Rodrigue Desmorat, Boris

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The modern assertion that a physics is a group [START_REF] Souriau | Grammaire de la Nature[END_REF] has important consequences regarding the invariance of the physical quantities. Hence, these questions have to be formulated and studied within the Mathematical framework of groups and Representation Theory [START_REF] Sternberg | Group theory and physics[END_REF]. In d-dimensional solid mechanics, the invariance properties of constitutive laws are formulated with respect to the full orthogonal group O(d). Quantities that are invariant with respect to the special orthogonal group SO(d) are called hemitropic invariants, while those that are invariant with respect to the full group O(d) are called isotropic invariants.

In continuum mechanics, constitutive equations, linear or non-linear, are naturally described using tensors [START_REF] Gurtin | The linear theory of elasticity[END_REF][START_REF] Truesdell | The non-linear field theories of mechanics[END_REF]. Yet classical in linear theories, constitutive tensors are also encountered in non-linear mechanics of materials such as, for instance, anisotropic elasto-plasticity (e.g. Hill yield tensor [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF]), in continuum damage mechanics (for a description of damage anisotropy see [START_REF] Chaboche | Description thermodynamique et phénoménologique de la viscoplasticité cyclique avec endommagement[END_REF][START_REF] Chaboche | Anisotropic creep damage in the framework of continuum damage mechanics[END_REF][START_REF] Onat | Effective properties of elastic materials that contain penny shaped voids[END_REF][START_REF] Lemaitre | Mécanique des matériaux solides. Dunod, english translation 1990 'Mechanics of Solid Materials[END_REF][START_REF] Ju | On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects[END_REF][START_REF] Lemaitre | Engineering Damage Mechanics[END_REF]) and in nonlinear piezoelectricity/magnetism (e.g. the magnetostriction morphic tensor [START_REF] Trémolet De Lacheisserie | Magnetostriction : theory and applications of magnetoelasticity[END_REF][START_REF] Hubert | Multiscale magneto-elastic modeling of magnetic materials including isotropic second order stress effect[END_REF]).

Orders of constitutive tensors, which are usually lower than four in classical linear elasticity and piezoelectricity, can however reach six in generalized continuum theories, such as strain gradient or micromorphic continua [START_REF] Auffray | Matrix representations for 3d strain-gradient elasticity[END_REF][START_REF] Auffray | Complete symmetry classification and compact matrix representations for 3d strain gradient elasticity[END_REF][START_REF] Polizzotto | Anisotropy in strain gradient elasticity: Simplified models with different forms of internal length and moduli tensors[END_REF][START_REF] Bertram | Compendium on gradient materials[END_REF]. They can be odd orders, forbidding then the definition of spectral invariants from Kelvin's matrix representation 1 [START_REF] Betten | Irreducible invariants of fourth-order tensors[END_REF][START_REF] Cowin | Eigentensors of linear anisotropic elastic materials[END_REF][START_REF] Kowalczyk-Gajewska | Review on spectral decomposition of hooke's tensor for all symmetry groups of linear elastic material[END_REF][START_REF] Bóna | Space of SO(3)-orbits of elasticity tensors[END_REF]. They can even rise up beyond order six when fabric tensors are involved [START_REF] Oda | Fabric tensor for discontinuous geological materials[END_REF][START_REF] Kanatani | Distribution of directional data and fabric tensors[END_REF][START_REF] Jemiolo | Fabric tensors in bone mechanics[END_REF][START_REF] Kachanov | Elastic solids with many cracks and related problems[END_REF][START_REF] Tikhomirov | On three-dimensional microcrack density distribution[END_REF][START_REF] Rahmoun | A 3d fourth order fabric tensor approach of anisotropy in granular media[END_REF][START_REF] Dormieux | Micromechanics of Fracture and Damage[END_REF][START_REF] Cormery | A stress-based macroscopic approach for microcracks unilateral effect[END_REF][START_REF] Caccuri | Tensorial nature of γ′-rafting evolution in nickel-based single crystal superalloys[END_REF]. A sound analysis of these tensors, of their invariants and their symmetry classes, gives precious information and modelling tools for the physics that can be described by them.

Tensors having symmetries can be described using their invariants and covariants [START_REF] Olive | Harmonic Factorization and Reconstruction of the Elasticity Tensor[END_REF][START_REF] Oliver-Leblond | Continuous anisotropic damage as a twin modelling of discrete bi-dimensional fracture[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]. Note however that the invariants must be chosen inside a given class: polynomial, rational or algebraic (such as the eigenvalues of a matrix). Besides, it is important to describe the tensors properties by a finite number of such invariants. To do so, mathematical definitions are required. For instance, one could be interested to describe the algebra of polynomial invariants using a minimal set of generators of this algebra (usually called an integrity basis), or to find a finite separating set of invariants -in a given class-which can be used to factor any invariant function of these tensors (set usually called a functional basis), with the property that an integrity basis is a functional basis [106]. If the later set is probably more pertinent in practical applications and of lower cardinality [START_REF] Ming | An irreducible polynomial functional basis of two-dimensional Eshelby tensors[END_REF][START_REF] Desmorat | Generic separating sets for three-dimensional elasticity tensors[END_REF][START_REF] Olive | Effective rationality of second-order symmetric tensor spaces[END_REF], there is however no algorithm to obtain such a set. On the contrary, there exist general -but complex-algorithms to compute a minimal integrity basis [START_REF] Weyl | The classical groups[END_REF][START_REF] Derksen | With two appendices by Vladimir L. Popov, and an[END_REF][START_REF] Olive | About gordan's algorithm for binary forms[END_REF].

The complexity increases with the order of the tensor. In 3D, this complexity is already high, while it remains reasonable in 2D. For example in 3D, a minimal integrity basis for the elasticity tensor is constituted of 294 invariants (for both SO [START_REF] Auffray | Complete symmetry classification and compact matrix representations for 3d strain gradient elasticity[END_REF] and O(3) [START_REF] Olive | A minimal integrity basis for the elasticity tensor[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]), while in 2D it is constituted of 5 invariants for O(2) and of 6 invariants for SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF]. This huge difference is roughly due to the following facts:

• The harmonic decomposition of the space of elasticity tensors contains more harmonic components in 3D than in 2D [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF][START_REF] Blinowski | Two-dimensional Hooke's tensors-isotropic decomposition, effective symmetry criteria[END_REF]. • The dimension of the space of harmonic tensors of order n is equal to 2n + 1 in 3D (like the space of spherical harmonics of degree n) and is equal to 2 in 2D (for n ≥ 1). The later is the dimension of the space generated by cos nθ and sin nθ in the Fourier decomposition.

In 3D, the first step consists in determining an integrity basis for harmonic tensors. Such results are available in the literature but only up to order 5 (i.e., up to degree 10 for binary forms [START_REF] Lercier | Covariant algebra of the binary nonic and the binary decimic[END_REF]), due to the exponential growth of the computations with the order n of the tensor. In the present contribution on the 2D case, we formulate a general method and provide an algorithm to compute a minimal integrity basis for isotropic/hemitropic invariants and covariants of a tensor or a family of tensors. In particular, we provide a minimal integrity bases for 2D totally symmetric tensors -such as fabric tensors [START_REF] Kanatani | Distribution of directional data and fabric tensors[END_REF]-up to order 12, which include the cases of 2D harmonic tensors of orders 2, 4, 6, 8, 10 and 12.

The structure of 2D constitutive tensors spaces and the determination of their symmetry classes has already been considered in a previous contribution [START_REF] Auffray | Handbook of bi-dimensional tensors: Part i: Harmonic decomposition and symmetry classes[END_REF]. Among the remaining open problems is the formulation of a systematic procedure to determine a minimum integrity basis 1 Moreover, the spectral invariants of the R d(d+1)/2 -Kelvin matrix of an elasticity tensor do not characterize the geometry of its orbits in R d (with d = 2 in 2D and d = 3 in 3D). They are algebraic invariants of O(d(d + 1)/2). Any function invariant with respect to O(d(d + 1)/2) is also invariant with respect to O(d), but the converse is false.

for either isotropic or hemitropic polynomial invariant functions on a given tensor space. Such a set is useful for two reasons; on the one hand, every polynomial invariant can be recast as a polynomial function of these generating invariants and, on the other hand, it separates the orbits (i.e. sets of tensors of the same kind), which means that at least one of the generating invariants takes different values, when evaluated on two tensors which are not of the same kind. This last property makes it possible to decide whether, or not, two tensors describe the same material up to an orthogonal transformation.

Let us first present a quick overview of this question in the mechanical community. The story started in 1946 with Weyl's pioneering book [START_REF] Weyl | The classical groups[END_REF], from which was extracted the methods and vocabulary still used in continuum mechanics nowadays. It took, however, a few decades before some basic results, such as the determination of a minimal integrity basis for a n-uplet of three-dimensional second-order tensors and vectors [START_REF] Spencer | Finite integrity bases for five or fewer symmetric 3 × 3 matrices[END_REF][START_REF] Spencer | Isotropic integrity bases for vectors and second-order tensors[END_REF][START_REF] Smith | On isotropic integrity bases[END_REF][START_REF] Smith | On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors[END_REF][START_REF] Kiral | On the constitutive relations for anisotropic materials -Triclinic, monoclinic, rhombic, tetragonal and hexagonal crystal systems[END_REF][START_REF] Rivlin | Collected Papers of R.S. Rivlin[END_REF], were published. In these approaches, a (non necessarily minimal) generating set is obtained first, using for instance Weyl's polarization theorem. In a second step, a reduction procedure is achieved, using polynomial relationships (syzygies) between the polynomial invariants [START_REF] Rivlin | Further Remarks on the Stress-Deformation Relation for Isotropic Materials[END_REF][START_REF] Smith | Constitutive Equations for Anisotropic and Isotropic Materials[END_REF], to eventually obtain a minimal integrity basis. For second-order tensors, these syzygies are essentially derived from Cayley-Hamilton's theorem, and are thus useless for higher order tensors. In that case, only partial results have been obtained [START_REF] Smith | The strain-energy function for anisotropic elastic materials[END_REF][START_REF] Smith | Isotropic invariants of traceless symmetric tensors of orders three and four[END_REF][START_REF] Betten | Irreducible invariants of fourth-order tensors[END_REF]107,[START_REF] Betten | Integrity bases for a fourth-rank tensor[END_REF]108], until recently.

In this approach [START_REF] Olive | Géométrie des espaces de tenseurs, une approche effective appliquée à la mécanique des milieux continus[END_REF], the problem of higher order tensors in 3D is recast in the realm of binary forms, which are complex homogeneous polynomials in two variables. Using a powerful tool called the Cartan map [START_REF] Cartan | The theory of spinors[END_REF][START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF][START_REF] Desmorat | Generic separating sets for three-dimensional elasticity tensors[END_REF], an integrity basis for the binary form of degree 2n can then be translated into an integrity basis for the harmonic tensor of degree n [START_REF] Desmorat | Generic separating sets for three-dimensional elasticity tensors[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]. The gain is that invariant theory of binary forms (also know as Classical Invariant Theory) is an area of mathematics which has been extensively studied by a wide number of prestigious mathematicians such as Gordan or Hilbert and in which an impressive number of results has already been produced. Combining these results with the use of the harmonic decomposition [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF][START_REF] Spencer | A note on the decomposition of tensors into traceless symmetric tensors[END_REF], integrity bases for the third order totally symmetric tensor and for the fourth-order elasticity tensor have been obtained recently [START_REF] Olive | Isotropic invariants of a completely symmetric third-order tensor[END_REF][START_REF] Olive | Géométrie des espaces de tenseurs, une approche effective appliquée à la mécanique des milieux continus[END_REF][START_REF] Olive | A minimal integrity basis for the elasticity tensor[END_REF]. In this approach, Gordan's algorithm for binary forms [START_REF] Gordan | Beweis, dass jede Covariante und Invariante einer Bineren Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist[END_REF][START_REF] Gordan | Ueber die Auflosung linearen Gleidungen mit reallen Coefficienten[END_REF][START_REF] Gordan | Uber das Formensystem Binaerer Formen[END_REF][START_REF] Gordan | Vorlesungen über Invariantentheorie[END_REF] is used first to generate a (non necessarily minimal) integrity basis and then, a reduction process using modern computational means is achieved to obtain minimality [START_REF] Olive | About Gordan's algorithm for binary forms[END_REF].

In 2D, the situation is much simpler and integrity bases are known in specific situations. For instance, regarding its practical importance for plate theory and laminated structures, the bidimensional (plane) elasticity tensor has been widely studied [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF][START_REF] Blinowski | Two-dimensional Hooke's tensors-isotropic decomposition, effective symmetry criteria[END_REF][START_REF] He | On the symmetries of 2D elastic and hyperelastic tensors[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] Vannucci | Stiffness design of laminates using the polar method[END_REF][START_REF] Vannucci | Plane anisotropy by the polar method[END_REF][START_REF] De Saxcé | Invariant measures of the lack of symmetry with respect to the symmetry groups of 2D elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF][START_REF] Desmorat | Tensorial polar decomposition of 2D fourth-order tensors[END_REF]. In [START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF], a comparative review of the literature on O(2) and SO(2) invariants of the elasticity tensor is provided. Recently, there has been an attempt (unfortunately with mistakes) to determine an integrity basis for fourth-order tensors of Eshelby typei.e. photoelasticity type [START_REF] Ming | An irreducible polynomial functional basis of two-dimensional Eshelby tensors[END_REF] -and partial results for the piezoelectricity tensor are already known [START_REF] Vannucci | The polar analysis of a third order piezoelectricity-like plane tensor[END_REF]. Nevertheless, analysing the literature, it appears that a general, effective and systematic method to compute a minimal integrity basis for coupled constitutive laws and more generally for tensors of any order is still lacking, and that almost no results are known for the covariant integrity bases [START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]. We moreover point out that the literature results on 2D invariants are rarely expressed using tensorial expressions.

Concerning practical applications, an integrity basis is required to formulate invariant relations that characterize intrinsic properties of a constitutive law, such as the belonging to a symmetry class [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF][START_REF] Auffray | Invariant-based reconstruction of bidimensional elasticity tensors[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF], the special (r 0 ) orthotropy [START_REF] Vannucci | Stiffness design of laminates using the polar method[END_REF][START_REF] Vannucci | Plane anisotropy by the polar method[END_REF] or the existence of a pentamode [START_REF] Milton | Which elasticity tensors are realizable[END_REF][START_REF] Kadic | On the practicability of pentamode mechanical metamaterials[END_REF][START_REF] Desmorat | Space of 2D elastic materials: a geometric journey[END_REF]. Such kind of relations are interesting for optimal design algorithms since they allow to formulate frame-independent constraints on the sought material [START_REF] Vincenti | Optimization of laminated composites by using genetic algorithm and the polar description of plane anisotropy[END_REF][START_REF] Ranaivomiarana | Concurrent optimization of material spatial distribution and material anisotropy repartition for two-dimensional structures[END_REF]. Invariants for higher order tensors will naturally find applications to extend this approach to the generalized (Mindlin) elasticity models used to describe the effective behaviour of architectured materials [START_REF] Abdoul-Anziz | Strain gradient and generalized continua obtained by homogenizing frame lattices[END_REF][START_REF] Poncelet | An experimental evidence of the failure of cauchy elasticity for the overall modeling of a non-centro-symmetric lattice under static loading[END_REF][START_REF] Rosi | Continuum modelling of frequency dependent acoustic beam focussing and steering in hexagonal lattices[END_REF].

The goal of the present contribution is to propose a general and effective method to compute a minimal integrity basis for any O(2) or SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] representation and to apply it to continuum mechanics constitutive laws. To this end, we follow the path traced by Vianello [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF] for the bidimensional elasticity tensor and formulate the problem within the framework of Invariant Theory [START_REF] Weyl | The classical groups[END_REF][START_REF] Springer | On the invariant theory of SU2[END_REF][START_REF] Olver | Classical invariant theory[END_REF][START_REF] Kraft | Classical Invariant Theory, a Primer[END_REF][START_REF] Derksen | With two appendices by Vladimir L. Popov, and an[END_REF]. This allows us to produce a minimal integrity basis for 2D higher order tensors with or without any particular index symmetry, under both groups O(2) and SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF]. Minimality of the integrity bases is obtained using a Computer Algebra System and an algorithm which is explained in details. Integrity bases are first formulated using complex variables (as in [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF]). In a more mechanistic way, we provide an original process allowing to translate all these expressions into tensorial ones.

This paper intents to be as self-contained as possible, and many illustrating examples are provided along the lines.

The outline of the paper is the following. In section 2, we recall basic operations on tensors, some of which are well-known and others are new. The link between totaly symmetric tensors and homogeneous polynomials is explained. Main concepts from the theory of linear representations of the orthogonal groups O(2) and SO(2) are presented in section 3. In section 4, we introduce basic notions of Invariant Theory, such as polynomial invariants and covariants. The main results of the paper are given in section 5, where integrity bases for bidimensional tensors of any order (and more generally any linear representation of O(2) and SO(2)) are derived. Hemitropic (i.e. for SO(2)) integrity bases produced are already minimal but isotropic ones (i.e. for O(2)) are not. A cleaning algorithm to achieve this task is formulated in section 6. The computed integrity bases are written in terms of complex monomials. It is more useful, in mechanics, to express them using tensorial operations. These translation rules are formulated in section 7.

In section 8, we illustrate the power of our methods by providing minimal integrity bases for an extensive list of constitutive tensors (up to twelfth-order) and coupled laws in mechanics of materials, including Eshelby/photoelasticity tensors, linear viscoelasticity, Hill elasto-plasticity, linear piezoelectricity and fabric tensors. Besides, four appendices are provided to detail and deepen some technical points.

Tensorial operations

This paper is about tensors polynomial invariants. In this section we recall basic operations on tensors, some of them are well-known, others are less. We shall denote by T n (R 2 ) = ⊗ n R 2 , the vector space of 2D tensors of order n. Using the Euclidean structure of R 2 , we will not make any difference between covariant, contravariant or mixed tensors. We will encounter tensors with various index symmetries, among them tensors which are totally symmetric. The subspace of T n (R 2 ) of totally symmetric tensors will be denoted by S n (R 2 ). Given T ∈ T n (R 2 ), the total symmetrization (over all subscripts) of T, denoted by T s is a projector from T n (R 2 ) onto S n (R 2 ). The following tensorial operations will be used (see also [START_REF] Desmorat | Generic separating sets for three-dimensional elasticity tensors[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]).

(1) The symmetric tensor product between two tensors S 1 ∈ S n 1 (R 2 ) and S 2 ∈ S n 2 (R 2 ), defined as

(2.1) S 1 ⊙ S 2 := (S 1 ⊗ S 2 ) s ∈ S n 1 +n 2 ,
(2) The r-contraction of two tensors

T 1 ∈ T n 1 (R 2 ) and T 2 ∈ T n 2 (R 2 )
, defined in any orthonormal basis as

(2.2) (T 1 (r) • T 2 ) i 1 •••i n 1 -r j r+1 •••jn 2 := T 1 i 1 •••i n 1 -r k 1 •••kr T 2 k 1 •••krj r+1 •••jn 2 ,
which is a tensor of order n 1 + n 2 -2r. (3) The skew-symmetric contraction between two totally symmetric tensors

S 1 ∈ S n 1 (R 2 )
and S 2 ∈ S n 2 (R 2 ) is defined as

(2.3) (S 1 × S 2 ) := -(S 1 • ε ε ε • S 2 ) s ∈ S n 1 +n 2 -2 (R 2 ),
where ε ε ε is the 2D Levi-Civita tensor. In any orthonormal basis (e e e 1 , e e e 2 ), we get ε ij = det(e e e i , e e e j ) and

(S 1 × S 2 ) i 1 ...i n 1 +n 2 -2 = -ε jk S 1 ji 1 ...in 1 S 2 ki n 1 +1 ...i n 1 +n 2 -2 s .
There is a well-known correspondence φ : S n (R 2 ) → P n (R 2 ) between totally symmetric tensors of order n on R 2 and homogeneous polynomials of degree n in two variables. Given T ∈ T n (R 2 ) we associate to it the polynomial p = φ(T) in P n (R 2 ), where (2.4) p(x x x) = T(x x x, . . . , x x x).

In components, this writes

p(x x x) = T i 1 i 2 ...in x i 1 x i 2 . . . x in , where x x x = (x 1 , x 2 ) = (x, y).
Note that φ(T) = φ(T s ) and that when restricted to S n (R 2 ), this correspondence S → p is a bijection, the inverse operation being given by the polarization of p (see [START_REF] Weyl | The classical groups[END_REF][START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF][START_REF] Baerheim | Harmonic decomposition of the anisotropic elasticity tensor[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]). Making use of this correspondence, the three tensorial operations defined above are recast into polynomial operations as follows. Let p i := φ(S i ) for i = 1, 2, be the homogeneous polynomials associated with S i ∈ S n i (R 2 ). Then, (1) the symmetric tensor product S 1 ⊙ S 2 translates as p 1 p 2 ;

(2) the symmetrized r-contraction (S 1 (r)

• S 2 ) s translates as

(2.5) {p 1 , p 2 } r := (n 1 -r)!(n 2 -r)! n 1 !n 2 ! r k=0 r k ∂ r p 1 ∂x k ∂y r-k ∂ r p 2 ∂x k ∂y r-k ;
(3) The skew-symmetric contraction S 1 × S 2 translates as

(2.6) [p 1 , p 2 ] := - 1 n 1 n 2 det(∇p 1 , ∇p 2 ),
where ∇ denotes the gradient.

Real linear representations of 2D orthogonal groups

The full orthogonal group in dimension 2, denoted by O(2), is defined as the set of linear isometries of the canonical scalar product on R 2 . In the canonical basis (e e e 1 , e e e 2 ), this group is represented by the two-by-two matrices g which satisfies g t g = I. In particular, we have det g = ±1. The subset of matrices g such that det g = 1 is a subgroup of O(2), denoted by SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF], which is the rotation group of the Euclidean space R 2 , each rotation being represented by the matrix

(3.1) r θ = cos θ -sin θ sin θ cos θ .
The full orthogonal group is obtained from SO(2) by adding the reflection with respect to the horizontal axis

(3.2) σ := 1 0 0 -1 .
Besides, each element of O(2) can be written, either as r θ (if det g = 1) or σr θ (if det g = -1), and we have the relations σ 2 = id, σr θ = r -θ σ, where σr θ = r -θ σ is the reflection with respect to the axis r -θ/2 (e e e 1 ) = cos θ 2 e e e 1sin θ 2 e e e 2 .

Next, we recall a few basic concepts in representation theory of groups. More details can be found, for instance, in [START_REF] Sternberg | Group theory and physics[END_REF]. Definition 3.1. A linear representation (V, ρ) of a group G on a vector space V is a linear action of G on V. More precisely, it is given by a mapping

ρ : G → GL(V),
where GL(V) is the group of invertible linear mappings on V and such that ρ(g 1 g 2 ) = ρ(g 1 )ρ(g 2 ), for all g 1 , g 2 ∈ G.

Linear representations of G = SO(2) and G = O(2) play a fundamental role in 2D solid mechanics. They arise, for instance, when V = Ela is the space of fourth-order plane elasticity tensors, or when V = Piez is the space of third-order bidimensional piezoelectric tensors.

A linear representation is by definition linear in v v v ∈ V and thus ρ(g) is represented by a matrix [ρ(g)] once a basis of V is fixed. A basic example is provided by the standard representation of O(2) on V = T n (R 2 ), the vector space of n-th order tensors of dimension 2 n . In the canonical basis of R 2 , (e e e 1 , e e e 2 ), the tensor ρ(g)T has for components

(3.3) (ρ(g)T) i 1 ...in = j 1 ,...,jn g i 1 j 1 • • • g injn T j 1 ...jn .
By the way, a natural basis for

T n (R 2 ) is provided by (e i 1 •••in )
, where

e i 1 •••in := e e e i 1 ⊗ • • • ⊗ e e e in ,
and the lexicographic order has been adopted on multi-index (i 1 , . . . , i n ). Thus, the corresponding matrix representation in this basis writes

[ρ(g)T] = [ρ(g)][T],
where, introducing the multi-index

I = (i 1 , . . . , i n ), J = (j 1 , . . . , j n ), [ρ(g)] IJ = g i 1 j 1 • • • g injn .
Three other examples are provided in Appendix A.

ϕ from V 1 to V 2 such that ϕ(ρ 1 (g)v v v 1 ) = ρ 2 (g)ϕ(v v v 1 ), for all v v v 1 ∈ V 1 and g ∈ G.
Real irreducible representations of 2D orthogonal groups are well-known (see for instance [START_REF] Golubitsky | Singularities and groups in bifurcation theory[END_REF]). Each real irreducible representation of SO(2) is either equivalent to the trivial representation on R, denoted by ρ 0 and defined by ρ 0 (g)λ = λ, for all g ∈ SO(2) and all λ ∈ R, or to the two-dimensional representation on R 2 given by

r θ ∈ SO(2) → ρ n (r θ ) = cos nθ -sin nθ sin nθ cos nθ ∈ GL(R 2 ),
and indexed by the integer n ≥ 1.

Each real irreducible representations of O(2) is either equivalent to the trivial representation on R, denoted by ρ 0 , the sign representation on R, denoted by ρ -1 and defined by

ρ -1 (g)ξ = (det g)ξ,
for all g ∈ O(2) and all ξ ∈ R (ξ is sometimes called a pseudo-scalar), or to one of the following representations on R 2 given by

ρ n (r θ ) = cos nθ -
ϕ := r π/2 : R 2 → R 2 ,
one can check that ϕ is an equivariant isomorphism:

ϕ • ρ n (r θ ) = ρn (r θ ) • ϕ, and ϕ • ρ n (σr θ ) = ρn (σr θ ) • ϕ.
For n = 0, we have ρ0 = ρ -1 , which is not equivalent to ρ 0 . In other words, there are neither pseudo-vectors and nor pseudo-tensors in 2D but there exists pseudo-scalars.

There are two models, useful in practice, for 2-dimensional irreducible representations of the orthogonal groups:

(1) The spaces H n (R 2 ) of homogeneous harmonic polynomials (polynomials with vanishing Laplacian) in two variables x, y of degree n ≥ 1, (2) The spaces H n (R 2 ) of nth-order harmonic tensors (totally symmetric tensors with vanishing traces). And, to complete these alternative models for n = 0 and n = -1, we set

H 0 (R 2 ) = H 0 (R 2 ) = R, with the trivial representation,
and

H -1 (R 2 ) = H -1 (R 2 ) = R
, with the sign representation. Any linear representation (V, ρ) of SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] or O(2) can be decomposed into a direct sum of irreducible representations. This is known as the harmonic decomposition of V and means that

(3.4) V ≃ H n 1 (R 2 ) ⊕ • • • ⊕ H np (R 2 ),
where n i ∈ {-1, 0, 1, 2, . . . } and where multiplicities are allowed. An explicit method to achieve such a decomposition for any representation V of SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] or O(2), based on the infinitesimal action of SO(2), is described in Appendix A. It extends, somehow, a method used in [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] Vannucci | Plane anisotropy by the polar method[END_REF][START_REF] Vannucci | The polar analysis of a third order piezoelectricity-like plane tensor[END_REF] for bidimensional elasticity (see also [START_REF] Spencer | A note on the decomposition of tensors into traceless symmetric tensors[END_REF][START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF] and [START_REF] Cowin | Properties of the anisotropic elasticity tensor[END_REF][START_REF] Baerheim | Harmonic decomposition of the anisotropic elasticity tensor[END_REF] for 3D elasticity using different approaches). The harmonic decomposition of the space of totally symmetric tensors is handled in Appendix B.

Example 3.5. The harmonic decomposition of fourth-order tensors under O(2) are given below, depending on their index symmetries, as described in [START_REF] Auffray | Handbook of bi-dimensional tensors: Part i: Harmonic decomposition and symmetry classes[END_REF], and where we have set T = (T ijkl ).

T ijkl No index symmetry 3H -1 (R 2 ) ⊕ 3H 0 (R 2 ) ⊕ 4H 2 (R 2 ) ⊕ H 4 (R 2 ) T (ij)kl One minor symmetry 2H -1 (R 2 ) ⊕ 2H 0 (R 2 ) ⊕ 3H 2 (R 2 ) ⊕ H 4 (R 2 )
T ij|kl Major symmetry

H -1 (R 2 ) ⊕ 3H 0 (R 2 ) ⊕ 2H 2 (R 2 ) ⊕ H 4 (R 2 ) T (ij)(kl) Minor symmetries H -1 (R 2 ) ⊕ 2H 0 (R 2 ) ⊕ 2H 2 (R 2 ) ⊕ H 4 (R 2 ) T ijkl = T jilk = T klij Normal Klein sym. 3H 0 (R 2 ) ⊕ H 2 (R 2 ) ⊕ H 4 (R 2 ) T (ijk)l Tot. sym. over 3 index H -1 (R 2 ) ⊕ H 0 (R 2 ) ⊕ 2H 2 (R 2 ) ⊕ H 4 (R 2 ) T (ij)|(kl) Elasticity 2H 0 (R 2 ) ⊕ H 2 (R 2 ) ⊕ H 4 (R 2 ) T (ijkl) Totally symmetric H 0 (R 2 ) ⊕ H 2 (R 2 ) ⊕ H 4 (R 2 )
Remark 3.6. Once an explicit harmonic decomposition has been fixed, a given tensor is parameterized by scalars, pseudo-scalars and harmonic tensors of order n ≥ 1 (which depend on two parameters). For instance, in the case of the elasticity tensor, we get

C = (λ, µ, h, H) where λ, µ ∈ H 0 (R 2 ), h ∈ H 2 (R 2 ), H ∈ H 4 (R 2 )
. The remarkable fact is that all the results we present in section 8 are independent of this choice. All formulas are valid, independently of the particular choice of an explicit harmonic decomposition.

Invariant theory in 2D

Let (V, ρ) be a linear representation of G = O(2) or G = SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF]. The action of G on V induces a linear representation of G on the algebra R[V] of polynomials functions on V, which will be denoted by ⋆, and which is given by

(4.1) (g ⋆ p)(v v v) := p(ρ(g) -1 v v v).
4.1. Invariant algebra. The invariant algebra of V under the group G, denoted by Inv(V, G) (and more usually by R[V] G in the Mathematical community), is defined as

Inv(V, G) := {p ∈ R[V], g ⋆ p = p, ∀g ∈ G} .
It is a subalgebra of R[V], which is furthermore finitely generated, thanks to Hilbert's theorem [START_REF] Hilbert | Theory of algebraic invariants[END_REF][START_REF] Sturmfels | Algorithms in Invariant Theory[END_REF]. Moreover, since the group action on polynomials preserves vector spaces of homogeneous polynomials of given degrees, it can always be generated by homogeneous polynomial invariants.

Definition 4.1 (Integrity basis). A finite set of G-invariant homogeneous polynomials {J 1 , . . . , J N } over V is a generating set (also called an integrity basis) of the invariant algebra Inv(V, G) if any G-invariant polynomial J over V is a polynomial function in J 1 , . . . , J N , i.e if J can be written as

J(v v v) = P (J 1 (v v v), . . . , J N (v v v)), v v v ∈ V
, where P is a polynomial function in N variables. An integrity basis is minimal if no proper subset of it is an integrity basis.

Remark 4.2. A minimal integrity basis of homogeneous invariants is not unique, several choices are possible but its cardinality, as well as the degree of the generators are independent of the choice of a particular basis [START_REF] Dixmier | Le nombre minimum d'invariants fondamentaux pour les formes binaires de degré 7[END_REF]. Definition 4.3. An homogeneous polynomial invariant is called reducible if it can be written as the product of two (non constant) homogeneous polynomial invariants, or more generally as a sum of products of two (non constant) homogeneous polynomial invariants. Otherwise, it is called irreducible. Lemma 4.4. Let J := {J 1 , . . . , J N } be a set of homogeneous polynomial invariants which generates Inv(V, G). If some J r ∈ J is reducible, then J \ {J r } is still a generating set of Inv(V, G).

Proof. Suppose that J r ∈ J is reducible. Then, it can be written as a sum of products of two (non constant) homogeneous polynomial invariants. J r = p,q

I p I q ,
where deg(I p I q ) = deg J r , for each pair (p, q). Thus, for each k, deg I k < deg J r (because I k is not constant). Besides, each I k writes as

I k = P k (J 1 , . . . , J N ).
But P k cannot depends on J r since deg I k < deg J r . The conclusion follows, since each I k , and thus J r , can then be rewritten as polynomial functions of the homogeneous invariants in J \ {J r }.

Corollary 4.5. A minimal integrity basis constituted of homogeneous invariants contains only irreducible invariants.

Covariant algebra.

There is a useful extension of the concept of invariant which is called a covariant [START_REF] Kraft | Classical Invariant Theory, a Primer[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF]. Its definition involves two representations ρ V and ρ W of G of the same group G (see definition 3.1).

Definition 4.6. A mapping v v v → c(v v v) from V to W is a covariant of v v v ∈ V of type W, if c(ρ V (g)v v v) = ρ W (g)c(v v v), ∀g ∈ G. It is called a polynomial covariant of type W if moreover the mapping v v v → c(v v v) is polynomial in v v v.
Remark 4.7. In the situations we are usually concerned with in mechanics, V and W are tensor spaces and the condition c(ρ

V (g)v v v) = ρ W (g)c(v v v) is written generally as c(g ⋆ T) = g ⋆ c(T),
where it is understood that the action ⋆ is the usual action of G on tensors. For G = SO(2), it simply means that the covariant c(T) of T is rotated by g if T is rotated by g. The concept of polynomial covariant is particularly useful when restricted to covariants of type S n (R 2 ) endowed with the tensorial representation (ρ n (g)S)(x x x 1 , . . . , x x x n ) = S(g -1 x x x 1 , . . . , g -1 x x x n ), g ∈ O(2).

In that case, to each polynomial covariant c of v v v of type S n (R 2 ), corresponds an homogeneous polynomial φ(c(v v v)) of degree n (see section 2), and

p c (v v v, x x x) := φ(c(v v v))(x x x) = c(v v v)(x x x, . . . , x x x)
is a polynomial function of both v v v and x x x. Moreover, we get

p c (ρ V (g)v v v, gx x x) = c(ρ V (g)v v v)(gx x x, . . . , gx x x) = (ρ n (g)c(v v v))(gx x x, . . . , gx x x) = c(v v v)(g -1 (gx x x), . . . , g -1 (gx x x)) = c(v v v)(x x x, . . . , x x x) = p c (v v v, x x x).
Therefore, to every covariant c of type S n (R 2 ), corresponds a unique invariant polynomial p c of (v v v, x x x). In other words, the polynomial covariants of type S n (R 2 ) can be identified with elements of the invariant algebra

R[V ⊕ R 2 ] G .
This justifies the following definition. Definition 4.9. The covariant algebra of V, i.e. the algebra generated by the polynomial G-covariants of V of type S n (R 2 ), denoted by Cov(V, G), is defined as

Cov(V, G) := Inv(V ⊕ R 2 , G), where G acts on V ⊕ R 2 as (v v v, x x x) → (ρ(g)v v v, gx x x), v ∈ V, x x x ∈ R 2 , g ∈ G.
Remark 4.10. Polynomial covariants appear to be much more useful than polynomial invariants to solve many problems in Invariant Theory such as the characterization of symmetry classes for instance [START_REF] Auffray | Invariant-based reconstruction of bidimensional elasticity tensors[END_REF][START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF] or the characterization of geometric properties [START_REF] Sturmfels | Algorithms in Invariant Theory[END_REF].

The covariant algebra Cov(V, G) is naturally bi-graded, by the degree d in v v v ∈ V, on one hand, called the degree of the covariant, and by the degree o in x x x = (x, y) ∈ R 2 , on second hand, called the order of the covariant. The set of covariants of order 0 is a subalgebra of Cov(V, G) which corresponds exactly to Inv(V, G) (this justifies the fact that the covariant algebra is an extension of the invariant algebra and contains more information). The set of covariants of degree d and order o is a finite dimensional vector subspace of Cov(V, G) which is denoted by

Cov d,o (V, G). Example 4.11. The following expressions are SO(2)-covariants of a ∈ S 2 (R 2 ) of respective type S 2 (R 2 ), S 2 (R 2 ) and S 6 (R 2 ): a 2 , 1 × a, a ⊙ a ⊙ a 3
, where the notation a n+1 = a n • a has been used. There polynomial counterparts write

a 2 (x x x, x x x) = x x x • a 2 • x x x, (1 × a)(x x x, x x x) = det(a • x x x, x x x), (x x x • a • x x x) 2 (x x x • a 3 • x x x),
and belong respectively to

Cov 2,2 (S 2 (R 2 ), SO(2)), Cov 1,2 (S 2 (R 2 ), SO(2)), Cov 5,6 (S 2 (R 2 ), SO (2)
).

The first one and the third one are also O(2)-covariants, but not the second one.

Computing integrity bases

The approach developed in this section has already been applied to plane elasticity by Vianello [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] Forte | A unified approach to invariants of plane elasticity tensors[END_REF], following a work of Pierce [START_REF] Pierce | Representations for transversely hemitropic and transversely isotropic stress-strain relations[END_REF], and in a related way by Verchery some years before [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF]. Our goal is to explain a systematic way to obtain integrity bases for the invariant algebra Inv(V, G) of any linear representation V of G = SO(2) or G = O(2), the computation of an integrity basis for Cov(V, G) being a particular case, since it is just the invariant algebra of V ⊕ R 2 . 5.1. SO(2) invariant algebras. We will start by studying the case of a representation V of the rotation group SO(2). To compute an integrity basis for V, the first step is to split V into irreducible components:

(5.1) V ≃ ν 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), ν 0 ∈ N, n k ∈ N * ,
where some n k may be equal (multiplicities of two-dimensional components H n k (R 2 ) are allowed). An explicit way to accomplish this task is detailed in Appendix A. Using this decomposition, a polynomial on V writes

p(λ 1 , . . . , λ ν 0 , a 1 , b 1 , . . . , a r , b r ),
where

λ j belongs to H 0 (R 2 ) = R and (a k , b k ) ∈ R 2 are the components of H k ∈ H n k (R 2
) in some basis.

Remark 5.1. Since each λ j is itself an invariant, every invariant polynomial which contains λ j , and which is not reduced to it, is necessarily reducible. Our goal being to compute a minimal integrity basis, and thus irreducible invariants of V, we can thus consider only invariant polynomials which depend on (a 1 , b 1 , . . . , a r , b r ).

Indeed, a minimal integrity basis for V consists of a minimal integrity basis of

H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ),
to which we must add λ 1 , . . . , λ ν 0 .

Following Vianello [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF], let us now introduce the complex variable z k := a k + ib k . Then, any real polynomial in (a 1 , b 1 , . . . , a r , b r ) can be recast as

α k ,β k c α 1 ,...,αr,β 1 ,...,βr z α 1 1 • • • z αr r z β 1 1 • • • z βr r , α i , β i ∈ N,
in which the condition of being real writes c β 1 ,...,βr,α 1 ,...,αr = c α 1 ,...,αr,β 1 ,...,βr ,

where c means the complex conjugate of c. The advantage of this choice of variables is that the action of SO(2) preserves the monomials, since

ρ(r θ )z k = e in k θ z k , ρ(r θ )z k = e -in k θ z k ,
and thus

r θ ⋆ z α 1 1 • • • z αr r z β 1 1 • • • z βr r = e i(n 1 (α 1 -β 1 )+•••+nr(αr -βr)) z α 1 1 • • • z αr r z β 1 1 • • • z βr r .
We need, therefore, only to compute invariant monomials.

Lemma 5.2. A monomial

(5.2) m := z α 1 1 • • • z αr r z β 1 1 • • • z βr r is SO(2)-invariant if and only if (α 1 , . . . , α r , β 1 , . . . , β r ) is solution of the linear Diophantine equation (5.3) n 1 α 1 + • • • + n r α r -n 1 β 1 -• • • -n r β r = 0.
A solution (α 1 , . . . , α r , β 1 , . . . , β r ) of (5.3) is called irreducible if it is not the sum of two nontrivial solutions, and reducible otherwise. It was shown by Gordan [START_REF] Gordan | Ueber die Auflosung linearen Gleidungen mit reallen Coefficienten[END_REF] (see also [START_REF] Kung | The invariant theory of binary forms[END_REF]Section 6.5] and [START_REF] Sturmfels | Algorithms in Invariant Theory[END_REF]Section 1.4]) that there is only a finite number of irreducible solutions of (5.3). Algorithms to compute these irreducible solutions can be found in [START_REF] Kryvyi | Algorithms for solving systems of linear diophantine equations in integer domains[END_REF][START_REF] Bruns | Normaliz: algorithms for affine monoids and rational cones[END_REF]. As one can expect, such minimal solutions lead directly to a minimal integrity basis of Inv(V, SO(2)).

Theorem 5.3. Let (V, ρ) be a real linear representation of SO(2) which decomposes as

(5.4) V ≃ ν 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), ν 0 ∈ N, n k ∈ N * .
Then, a minimal integrity basis of Inv(V, SO(2)) consists of the homogeneous invariants

(5.5) λ i , |z k | 2 , Re(m l ), Im(m l ),
where

1 ≤ i ≤ ν 0 , 1 ≤ k ≤ r
, and m l are the irreducible solutions of (5.3) such that m l = m l .

Proof. Consider first the algebra of complex invariant polynomials 2) .

A := C[z 1 , z 1 , . . . , z r , z r ] SO (
It follows from [START_REF] Sturmfels | Algorithms in Invariant Theory[END_REF]Lemma 1.4.2], that a minimal integrity basis for A is given by

z k z k , m l , m l ,
where 1 ≤ k ≤ r and m l are the irreducible solutions of ( 5.3) such that m l = m l . Thus, A is also generated by

z k z k , Re(m l ) = 1 2 (m l + m l ), Im(m l ) = 1 2i (m l -m l ),
which is still minimal. Now every real polynomial in

Inv(V, SO(2)) = R[a 1 , b 1 , . . . , a r , b r ] SO(2)
is a real polynomial in

z k z k , Re(m l ) = 1 2 (m l + m l ), Im(m l ) = 1 2i (m l -m l ).
Hence, this set is a generating set of R[a 1 , b 1 , . . . , a r , b r ] SO (2) which is also minimal. Otherwise, one of these invariants could be written as a real polynomial in the others and this would contradict the fact that this set is minimal as a generating set of A. As already stated (see remark 5.1), we conclude that

λ i , |z k | 2 , Re(m l ), Im(m l ),
is a minimal integrity basis of Inv(V, SO(2)).

Example 5.4. A minimal integrity basis for the action of SO(2) on Ela has been computed in [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF], using the harmonic decomposition, and representing an elasticity tensor C = (λ, µ, h, H) (see remark 3.6) in complex form. The basis writes

λ, µ, |z 2 | 2 , |z 4 | 2 , Re(z 2 2 z 4 ), Im(z 2 2 z 4 ), where z 2 = h 11 + ih 12 and z 4 = H 1111 + iH 1112 are the components of H 2 := h ∈ H 2 (R 2 ) and H 4 := H ∈ H 4 (R 2 ) in some orthonormal basis of R 2 .
The tensorial expressions of these invariants are provided in example 7.5.

O(2) invariant algebras. Consider now a representation V of the orthogonal group O(2), which decomposes as

V ≃ m -1 H -1 (R 2 ) ⊕ m 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ),
where m -1 ≥ 0, m 0 ≥ 0 and n i ≥ 1. An integrity basis of Inv(V, O(2)) will be obtained from one of the invariant algebra of the restriction of the representation of O(2) on V to its subgroup SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF]. This integrity basis will not be minimal in general and further computations will be necessary to extract from it a minimal integrity basis. Theorem 5.5. Let (V, ρ) be a real linear representation of O(2) which decomposes as

V ≃ m -1 H -1 (R 2 ) ⊕ m 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ),
where m -1 , m 0 ∈ N and n k ∈ N * . Then, an integrity basis for Inv(V, O(2)) consists of the homogeneous invariants

(5.6) λ k , |z l | 2 , Re(m l ), ξ i ξ j , ξ i Im(m l ), Im(m p ) Im(m q ),
where

λ k ∈ H 0 (R 2 ), ξ i ∈ H -1 (R 2 )
and where m l are the irreducible solutions of (5.3) such that m l = m l and only remains the terms Im(m p ) Im(m q ) for which neither m p m q , nor m p m q contains a factor z s z s for some s ∈ {1, . . . , r}.

The proof of theorem 5.5 requires a useful tool in invariant theory called the Reynolds operator, which is defined as follows.

Definition 5.6. Given a compact group G and a linear representation V of G, the Reynolds operator is the linear projector from R[V] onto the invariant algebra Inv(V, G), defined as

(5.7) R G (p) := G (g ⋆ p) dµ, p ∈ R[V],
where dµ is the Haar measure on G.

Definition 5.7. Given a compact group G, the Haar measure is a (bi-invariant) probability measure on G and is uniquely defined [START_REF] Sternberg | Group theory and physics[END_REF]. For G = SO(2), it writes as

SO(2) f (g) dµ = 1 2π 2π 0 f (r θ ) dθ,
for every continuous function f on SO(2), whereas for G = O(2), it writes as

O(2) f (g) dµ = 1 4π 2π 0 f (r θ ) dθ + 1 4π 2π 0 f (σr θ ) dθ,
for every continuous function f on O(2).

Proof of theorem 5.5. Consider an O(2)-invariant polynomial p. It is obviously invariant under SO(2) and we get thus

(5.8) p = R O(2) (p) = 1 2 R SO(2) (p) + σ ⋆ R SO(2) (p) = 1 2 (p + σ ⋆ p).
Now as an element of Inv(V, SO(2)) and using theorem 5.3, p can be written as a polynomial expression p = P (λ k , ξ i , |z l | 2 , Re(m l ), Im(m l )), and we have moreover

σ ⋆ λ k = λ k , σ ⋆ ξ i = -ξ i , and σ ⋆ |z l | 2 = |z l | 2 , σ ⋆ Re(m l ) = Re(m l ), σ ⋆ Im(m l ) = -Im(m l ). We get thus σ ⋆ p = P (λ k , -ξ i , |z l | 2 , Re(m l ), -Im(m l )
). Now, using (5.8), we have

p = 1 2 P (λ k , ξ i , |z l | 2 , Re(m l ), Im(m l )) + P (λ k , -ξ i , |z l | 2 , Re(m l ), -Im(m l )) ,
and expanding P , we deduce thereby that Inv(V, O(2)) is generated by the homogeneous invariants

λ k , |z l | 2 , Re(m l ), ξ i ξ j , ξ i Im(m l ), Im(m p ) Im(m q ). Note however that Im(m p ) Im(m q ) = Re(m p ) Re(m q ) -Re(m p m q ) = Re(m p m q ) -Re(m p ) Re(m q ).
Hence, we can remove Im(m p ) Im(m q ) from the list of generators, each time m p m q or m p m q can be recast as (z s z s )m, for some s ∈ {1, . . . , r} and m is a monomial which satisfies (5.3). Indeed, then Im(m p ) Im(m q ) is reducible and can be removed from the set of generators by lemma 4.4. This applies, in particular, to each invariant (Im(m l )) 2 .

The elimination of Im(m p ) Im(m q ), each time m p m q or m p m q contains a factor z s z s in the list of generators, does not lead, in general, to a minimal basis, even if it reduces a priori the number of generators, sometimes drastically.

Remark 5.8. For those of you who have been involved in similar calculations, the problem of whether such invariants as products Im m p Im m q could always be removed a priori from a minimal basis of Inv(V, O(2)) founds here a definitive answer. Indeed, there are examples in section 8 where such products cannot be removed (even in the case of totally symmetric tensors, see subsection 8.4, for instance).

A reduction procedure, which we call cleaning and described in section 6 is thus required to obtain a minimal integrity basis or to check that a given basis is already minimal. In practice, and the argument will be used when applying the cleaning procedure, it is enough to reduce the integrity basis (5.9)

B := |z k | 2 , Re(m l ), Im(m i ) Im(m j ) of Inv(H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), O (2)
), to obtain a minimal integrity basis of the full space

m -1 H -1 (R 2 ) ⊕ m 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ).
The argument is formalized as the following theorem.

Theorem 5.9. Let MB be a minimal integrity basis of

Inv(H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), O(2)),
extracted from B (5.9). Then, a minimal integrity basis for

Inv(m -1 H -1 (R 2 ) ⊕ m 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), O(2)),
is given by MB ∪ {λ k , ξ i ξ j , ξ i Im(m l )} , where 0 ≤ i, j ≤ m -1 , 0 ≤ k ≤ m 0 , and m l are the irreducible solutions of (5.3) such that m l = m l Proof. The invariant algebra Inv(V, O(2)) is multi-graded; each invariant writes uniquely as a sum of invariants which are multi-homogeneous relatively to the decomposition

V ≃ m -1 H -1 (R 2 ) ⊕ m 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ).
In other words,

Inv(V, O(2)) = K Inv K (V, O(2)),
where

K := (s 1 , . . . s m -1 , e 1 , .
. . e m 0 , k 1 , . . . , k r ), is a multi-index in which s i indicates the degree in ξ i , e k indicates the degree in λ k , k i indicates the degree in H n i and Inv K (V) is the vector space of multi-homogeneous invariants of multi-degree K. Any relation among multi-homogeneous invariants happens in one vector space Inv K (V, O(2)). Thus, neither λ k , nor ξ i ξ j , can be recast using other invariants from the set (5.6). This is because the spaces Inv K (V, O(2)) which contains either λ k or ξ i ξ j are one-dimensional. This is also true for ξ i Im(m l ), not because the corresponding space Inv K (V, O(2)), to which it belongs is one-dimensional, but because if it could be recast using other invariants from the set (5.6), then Im(m l ) could be re-written using |z k | 2 , Re(m i ) and Im(m j ) (j = l), which would lead to a contradiction.

Example 5.10. A minimal integrity basis for the action of O(2) on Ela has been computed in [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF]. It consists in the following five invariants λ, µ, z 2 z 2 , z 4 z 4 , Re(z 2 2 z 4 ), where z 2 = h 11 + ih 12 and z 4 = H 1111 + iH 1112 . The tensorial expressions of these invariants are provided in example 7.5.

We finally formulate as a theorem another reduction result, which avoids useless computations. Proof. Let W be a stable subspace of V. Since each Remark 5.12. Theorem 5.11 applies, in particular to any stable subspace W of V = T n (R 2 ), defined by some index symmetries, and thus in particular to W = S n (R 2 ). It applies also to the space W = S n-2k (R 2 ), which can be considered as a subspace of V = S n (R 2 ), because there is a natural and equivariant linear embedding

Theorem 5.11. Let V = H n 1 (R 2 ) ⊕ • • • ⊕ H np (R 2 ), where n k ≥ -1. Then, any stable subspace W of V writes W = H n k 1 (R 2 ) ⊕ • • • ⊕ H n kp (R 2 ), where n k 1 , . . . ,
H n (R 2 ) is an irreducible representation, each stable subspace W ∩ H n (R 2 ) of H n (R 2 ) is either {0} or H n (R 2
S n-2k (R 2 ) → S n (R 2 ), S → 1 ⊙ • • • ⊙ 1 k copies ⊙S.

Cleaning algorithm

Starting from a finite generating set B (5.9) of the invariant algebra (6.1)

A := Inv(H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), O(2)), n k i ≥ 1,
the cleaning algorithm produces a minimal integrity basis MB extracted from B. The invariant algebra A is multi-graded by the fact that each polynomial invariant can be uniquely decomposed into a sum of polynomial invariants which are homogeneous into each factor (multiplicities allowed)

H n 1 (R 2 ), . . . , H nr (R 2 ), of respective degrees k 1 , k 2 , . . . , k r
. This information will be encoded into the multi-index

K := (k 1 , k 2 , . . . , k r ).
Therefore, the invariant algebra A can be decomposed as the direct sum

A = K A K ,
where each A K is the finite dimensional subspace of A consisting of multi-homogeneous invariants of multi-degree K. The remarkable fact is that the dimension a K of A K can be computed a priori using the Hilbert series of A (see Appendix C), which writes, by theorem C.2 and remark C.3,

H(t 1 , . . . , t r ) = K a K t k 1 1 • • • t kr r , a K = 1 2 (b K + β K ),
where β K = 0 each time one of the k i is odd and β K = 1 otherwise, and where b K is the number of solutions (α 1 , . . . , α r ) of the linear Diophantine equation

(6.2) 2α 1 n 1 + • • • + 2α r n r = k 1 n 1 + • • • + k r n r , α i ≥ 0.
Let now B K := B ∩ A K be the subset of B of homogeneous polynomials with multi-degree K = (k 1 , k 2 , . . . , k r ). Choosing a total order on the set of multi-index K, leads to a partitioning of B as

B = N i=0 B K i , where K i ≺ K j , if i < j.
Remark 6.1. Any finite set S of p homogeneous polynomials in A K is a family of vectors v v v 1 , . . . , v v v p in the finite dimensional space A K of dimension a K and we can thus define its rank, Rank(S).

The proposed cleaning algorithm with

• inputs: B K i , a K i , with K i ≺ K j , for all i < j,
• output : MB, consists in:

(1) Initialization : determine a subfamily F 0 ⊂ B K 0 of linearly independent polynomials such that Rank(F 0 ) = a K 0 . (2) Iteration step n (1 ≤ n ≤ N ): suppose that we have obtained, at step n -1, the family F n-1 = {I 1 , . . . , I s } and note that F n-1 may contain homogeneous polynomials with different multi-indices K(I 1 ), . . . , K(I s ) but all are strictly lower than K n , where K(I) stands for the multi-index of homogeneous polynomial I.

(a) Determine the finite set R Kn of all reducible homogeneous polynomials of multidegree K n that can be constructed, in two steps, using elements of F n-1 : (i) Find the p solutions α j 1 , α j 2 . . . , α j s (1 ≤ j ≤ p) of the linear Diophantine system (6.3)

α 1 K(I 1 ) + . . . + α s K(I s ) = K n , (ii) If p > 0, R Kn = I α j 1 1 I α j 2 2 • • • I α j s s ; 1 ≤ j ≤ p , else R Kn = ∅, (b) if Rank(R Kn ) = a Kn , X Kn = ∅, go to (d), (c) Determine a subset X Kn ⊂ B Kn of minimal cardinal such that Rank(R Kn ∪ X Kn ) = a Kn ,
i.e. check one by one the invariants of B Kn that need to be added to match the dimension a Kn . This requires to compute the rank of the new set of vectors in A Kn , each time we add a new element, (d)

F n := F n-1 ∪ X Kn , (3) Termination: MB := F N .
The cleaning algorithm was applied with the following specifications.

• All minimal solutions of the Diophantine equation (5.3) were obtained using the software of algebraic geometry Normaliz [START_REF] Bruns | Normaliz: algorithms for affine monoids and rational cones[END_REF]; • The Diophantine equation (6.2) as well as the linear Diophantine system (6.3) were solved using Mathematica [START_REF] Inc | Mathematica[END_REF]; • The adopted total order on multi-index K = (k 1 , . . . , k r ) was the lexicographic order

K i K j ⇔ k i 1 = k j 1 , . . . , k i q = k j q , k i q+1 < k j q+1 ,
for some 1 ≤ q ≤ r -1;

• In step (2)(c) of the algorithm, it is necessary to order the element of B Kn = B ∩ A Kn to be tested. We have used the Mathematica built-in function Sort.

Remark 6.2. When the covariant algebra is involved, the cleaning algorithm is applied to the invariant algebra

Inv(R 2 ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), O(2)).
In that case, the multi-index has been numbered as

K := (k 0 , k 1 , k 2 , . . . , k r ),
where k 0 represents the degree in x x x, i.e. the order of the associated covariant. The choice of the adopted lexicographic order implies that the cleaning is processed by increasing the order of covariants, first. Of course, other total orders on the set of multi-index are possible and they may be more adapted to other situations considered.

From complex monomials to tensor covariants

Regarding mechanical applications, an integrity basis should be expressed in terms of tensors invariants, rather than in terms of complex monomials. A translation of the real and imaginary parts Re(m l ), Im(m l ) of the previous monomials is thus mandatory. As recalled in section 2, there is a natural correspondence

φ : S n (R 2 ) → P n (R 2 ),
which associates to any totally symmetric tensor S of order n, an homogeneous polynomial p of degree n, which writes as p(x x x) = S(x x x, . . . , x x x), x x x = (x, y).

Under this isomorphism, which is O(2)-equivariant, the subspace H n (R 2 ) of harmonic tensors of order n (traceless tensors) is sent to the subspace of homogeneous harmonic polynomials H n (R 2 ) (polynomials with vanishing Laplacian). A natural basis for H n (R 2 ) is given by the real and imaginary parts of the complex function

z n = (x + iy) n , p (n) 1 = Re(x + iy) n , p (n)
2 = Im(x + iy) n . This basis corresponds to the image under φ of the following basis of H n (R 2 ) (7.1) where ⊙ stands for the symmetric tensor product (2.1) and e e e p i := e e e i ⊙ • • • ⊙ e e e i means the tensor product of p copies of vector e e e i . Thus, any harmonic polynomial h in H n (R 2 ) writes h = ap

K (n) 1 = ⌊ n 2 ⌋ k=0 n 2k ( -
(n) 1 + bp (n)
2 , and the harmonic tensor

H = φ -1 (h) in H n (R 2 ) writes H = a K (n) 1 + b K (n) 2 , where a = H 11•••11 b = H 11•••12
while the other components [START_REF] Kanatani | Distribution of directional data and fabric tensors[END_REF] of H are

H 1•••1 n-2p 2•••2 2p = (-1) p H 1•••11 and H 1•••1 n-2p-1 2•••2 2p+1 = (-1) p H 1•••12 .
Remark 7.1. In both cases, the matrix form of ρ n in these bases is [ρ n (r θ )] = cos nθsin nθ sin nθ cos nθ .

Note however, that none of the defined bases are orthonormal for the natural scalar products on both spaces. There are however orthogonal and their norms are equal (see remark A.3). Normalizing the bases, will not change the matrix representation and is thus inessential.

Consider now a representation

(7.2) V ≃ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ), of G = SO(2) or G = O(2)
, where n k ≥ 1 for each k. Generating sets for Inv(V, G) have been provided in section 5, but in terms of monomials

m = z α 1 1 • • • z αr r z β 1 1 • • • z βr r
, whose exponents satisfy the linear Diophantine equation (5.2). Here, z k = a k + ib k corresponds to the components (a k , b k ) of the factor H k ∈ H n k (R 2 ) in the direct sum (7.2) relative to the basis (K

(n) 1 , K (n)
2 ), where we have used the correspondence

h k (x x x) = H k (x x x, . . . , x x x) ∈ H n k (R 2 ),
and recast h k as a polynomial function of the complex variables z = x + iy and z = xiy.

It is the goal of this section to translate real and imaginary parts of the monomials m into tensors invariants. To do so, observe first that

h k = Re(z k z n k ) = Re(z k z n k ) = 1 2 (z k z n k + z k z n k ) ,
and its conjugate harmonic function h k writes as

h k = Im(z k z n k ) = -Im(z k z n k ) = 1 2i (z k z n k -z k z n k ) ,
where z = x + iy and z k = a k + ib k . We will now provide three theorems which enable to translate real and imaginary parts of monomials m into tensors invariants. Their proofs are provided in Appendix D.

Theorem 7.2. Let h k = φ(H k ) ∈ H n k (R 2 )
, where n k ≥ 1. Then,

h k = Im(z k z n k ) = φ(1 × H k ).
Theorem 7.3. Let H j ∈ H n j (R 2 ) be harmonic tensors where n j ≥ 1 and set φ(H j ) = Re(z j z n j ).

Then Re z 1 • • • z p z n 1 +•••+np = 2 p-1 φ (H 1 ⊙ • • • ⊙ H p ) ′ .
Theorem 7.4. Let H j ∈ H n j (R 2 ) be harmonic tensors where n j ≥ 1 and set φ(H j ) = Re(z j z n j ).

Let

N 1 = n 1 + • • • + n p , N 2 = n p+1 + • • • + n p+s and assume that N 1 ≤ N 2 . Then Re z 1 • • • z p z p+1 • • • z p+s z N 2 -N 1 = 2 (p+s-1-N 1 ) φ (H 1 ⊙ • • • ⊙ H p ) ′ (N 1 ) • (H p+1 ⊙ • • • ⊙ H p+s ) ′ ,
and

Im z 1 • • • z p z p+1 • • • z p+s z N 2 -N 1 = 2 (p+s-2N 1 ) (N 1 + N 2 -2)! (N 1 -1)!(N 2 -1)! φ tr (N 1 -1) (H 1 ⊙ • • • ⊙ H p ) ′ × (H p+1 ⊙ • • • ⊙ H p+s ) ′ = 2 (p+s-1-N 1 ) φ (H 1 ⊙ • • • ⊙ H p ) ′ (N 1 ) • ([1 × H p+1 ] ⊙ • • • ⊙ H p+s ) ′ ] = -2 (p+s-1-N 1 ) φ ([1 × H 1 ] ⊙ • • • ⊙ H p ) ′ (N 1 ) • (H p+1 ⊙ • • • ⊙ H p+s ) ′ ] .
Example 7.5. Let C ∈ Ela be a bidimensional elasticity tensor. Its harmonic decomposition writes C ≃ (λ, µ, h, H), where h ∈ H 2 (R 2 ) and H ∈ H 4 (R 2 ) (see example 3.5 and remark 3.6). Writing φ(h) = Re(z 2 z 2 ), and φ(H) = Re(z 4 z 4 ), the translation of monomial invariants given in 5.4 and 5.10, namely

z 2 z 2 , z 4 z 4 , Re(z 2 2 z 4 ), Im(z 2 2 z 4 ) write z 2 z 2 = 1 2 (h (2)
• h) = 1 2 h : h,

z 4 z 4 = 1 2 3 (H (4)
•

H) = 1 8 H :: H, Re(z 2 2 z 4 ) = 1 4 ((h ⊙ h) ′ (4) • H) = 1 4 (h : H : h),
while there are several possibilities to translate Im(z 2 2 z 4 ):

Im(z 2 2 z 4 ) = 5 8 tr 3 ((h ⊙ h) ′ × H) = 1 4 (h ⊙ h) ′ (4) • (1 × H) = 1 4 h : (1 × H) : h = - 1 4 ((1 × h) ⊙ h) ′ (4) • H = - 1 4 h : H : (1 × h).
We deduce thus the following results.

(1) A minimal SO(2)-integrity basis for C ∈ Ela is λ, µ, h : h, H :: H, h : H : h, h : H : (1 × h).

(2) A minimal O(2)-integrity basis for C ∈ Ela is λ, µ, h : h, H :: H, h : H : h.

Minimal covariant bases for most common constitutive tensors and laws

In this section, we illustrate the power of the method explained in this paper by providing a minimal integrity basis for an exhaustive list of constitutive tensors and laws which involve several tensors. More precisely, applying theorems 5.3, 5.5, and 5.9, and using the cleaning algorithm detailed in section 6, we obtain explicit results in 2D for:

• Third-order tensors with no index symmetry T 3 (R 2 ), thus for third-order tensors with any kind of index symmetries (by theorem 5.11), and in particular for the piezoelectricity tensor P ∈ Piez; • Fourth-order tensors with no index symmetry T 4 (R 2 ), thus for fourth-order tensors with any kind of index symmetries (by example 3.5 and theorem 5.11), and in particular for the elasticity tensor C ∈ Ela and the photoelasticity/Eshelby tensor Π Π Π ∈ Gel; • The complex viscoelasticity tensor, or more precisely its de-complexification, Ela ⊕ Ela, • The Hill elasto-plasticity constitutive equations;

• The linear piezoelectricity constitutive law, which involves three constitutive tensors, the dielectric permittivity tensor (of order two), the piezoelectricity tensor (of order three) and the elasticity tensor (of order four); • Twelfth-order totally symmetric tensors S ∈ S 12 (R 2 ) and thus for totally symmetric fabric tensors [START_REF] Kanatani | Distribution of directional data and fabric tensors[END_REF] of order 4, 6, 8 and 10 (by remark 5.12).

In each case, we provide an harmonic decomposition and a minimal integrity basis of the covariant algebra (except for S 12 (R 2 ), for which we provide only an integrity basis for its invariant algebra due to its very large cardinal), and this both for G = O(2) and G = SO(2). Remark 8.1. For each produced SO(2)-integrity basis, the generators I satisfy either σ ⋆ I = I or σ ⋆ I = -I. In the first case, we shall refer to I as an isotropic invariant (since it is O(2)invariant), and in the second case, as an hemitropic invariant. Besides, the following notation has been adopted. For each factor H n (R 2 ) which occurs in the harmonic decomposition provided, the corresponding variable is written as z n , if there is only one component H n (R 2 ) in this decomposition or z na , z nb , z nc , . . . if the component H n (R 2 ) appears with multiplicity. 8.1. Third-order tensors. The harmonic decomposition of T 3 (R 2 ) is the same for SO(2) and O(2) and writes

T 3 (R 2 ) ≃ 3H 1 (R 2 ) ⊕ H 3 (R 2
). We will thus write T = (z 1a , z 1b , z 1c , z 3 ), after the choice of an explicit harmonic decomposition (such as example A.5) and we have the following result. Theorem 8.2. A minimal integrity basis for Cov(T 3 (R 2 ), SO(2)) consists in the 57 covariants (30 invariants) of Table 1 andTable 2

z] 46 1 1 Im[z 1b z] 47 1 1 Im[z 1c z] 48 1 3 Im[z 2 1a z 3 z] 49 1 3 Im[z 2 1b z 3 z] 50 1 3 Im[z 2 1c z 3 z] 51 1 3 Im[z 1a z 1b z 3 z] 52 1 3 Im[z 1a z 1c z 3 z] 53 1 3 Im[z 1b z 1c z 3 z] 54 2 2 Im[z 1a z 3 z 2 ] 55 2 2 Im[z 1b z 3 z 2 ] 56 2 2 Im[z 1c z 3 z 2 ] 57 3 1 Im[z 3 z 3 ]
An application of theorem 8.2 concerns the bidimensional piezoelectricity third-order tensor P (also denoted e in the IEEE Standard on Piezoelectricity, ANSI/IEEE 176 -1987), with index symmetry P ijk = P ikj . It relates the electric displacement D D D to the stress tensor σ σ σ ∈ S 2 (R 2 ), at vanishing electric field, as [START_REF] Royer | Elastic waves in solids I: Free and guided propagation[END_REF] D D D = P : σ σ σ, D i = P ijk σ jk .

The space of 2D piezoelectricity tensors, noted Piez, has the same harmonic decomposition for both O(2) and SO(2) and writes [START_REF] Geymonat | Symmetry classes of piezoelectric solids[END_REF] # order degree Formula

] 2H 1 (R 2 ) ⊕ H 3 (R 2
1 0 2 v v v • v v v 2 0 2 w w w • w w w 3 0 2 v v v • w w w 4 0 2 v v v × w w w 5 0 2 H . . . H 6 0 4 (v v v • H • v v v) • v v v 7 0 4 (v v v • H • v v v) • w w w 8 0 4 (w w w • H • w w w) • w w w 9 0 4 (w w w • H • w w w) • v v v 10 0 4 (v v v • H • v v v) • (1 × v v v) 11 0 4 (v v v • H • v v v) • (1 × w w w) 12 0 4 (w w w • H • w w w) • (1 × v v v) 13 0 4 (w w w • H • w w w) • (1 × w w w)
# order degree Formula

14 1 1 v v v 15 1 1 w w w 16 1 1 1 × v v v 17 1 1 1 × w w w 18 1 3 v v v • H • v v v 19 1 3 w w w • H • w w w 20 1 3 v v v • H • w w w 21 1 3 v v v • H • (1 × v v v) 22 1 3 w w w • H • (1 × w w w) 23 1 3 v v v • H • (1 × w w w) 24 2 0 1 25 2 2 H • v v v 26 2 2 H • w w w 27 2 2 H • (1 × v v v) 28 2 2 H • (1 × w w w) 29 3 1 H 30 3 1 1 × H Table 4. A minimal integrity basis for Cov(Piez, O (2))
# order degree Formula

1 0 2 v v v • v v v 2 0 2 w w w • w w w 3 0 2 v v v • w w w 4 0 2 H . . . H 5 0 4 (v v v • H • v v v) • v v v 6 0 4 (v v v • H • v v v) • w w w 7 0 4 (w w w • H • w w w) • w w w 8 0 4 (w w w • H • w w w) • v v v # order degree Formula 9 1 1 v v v 10 1 1 w w w 11 1 3 v v v • H • v v v 12 1 3 w w w • H • w w w 13 1 3 v v v • H • w w w 14 2 0 1 15 2 2 H • v v v 16 2 2 H • w w w 17 3 1 H
Finally, we will complete our investigations of third-order tensors, in order to be fully exhaustive, by adding the space of totally symmetric tensors S 3 (R 3 ). Its harmonic decomposition writes

S 3 (R 3 ) ≃ H 1 (R 2 ) ⊕ H 3 (R 2
= (v v v, H), with v v v ∈ H 1 (R 2 ) and H ∈ H 3 (R 2 )
, consists in the 13 covariants (4 invariants) of Table 5. A minimal integrity basis for Cov(S 3 (R 2 ), O(2)) consists in the 8 covariants (3 invariants) of Table 6.

8.2.

Fourth-order tensors. The harmonic decomposition of T 4 (R 2 ) relative to SO(2) writes

6H 0 (R 2 ) ⊕ 4H 2 (R 2 ) ⊕ H 4 (R 2 ).
We will thus write T = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , z 2a , z 2b , z 2c , z 2d , z 4 ), after the choice of an explicit harmonic decomposition. For O(2), we get

3H -1 (R 2 ) ⊕ 3H 0 (R 2 ) ⊕ 4H 2 (R 2 ) ⊕ H 4 (R 2 ),
and we will have T = (ξ 1 , ξ 2 , ξ 3 , λ 1 , λ 2 , λ 3 , z 2a , z 2b , z 2c , z 2d , z 4 ).

Table 5. A minimal integrity basis for Cov(S 3 (R 2 ), SO(2))

# order degree Formula

1 0 2 v v v • v v v 2 0 2 H . . . H 3 0 4 (v v v • H • v v v) • v v v 4 0 4 (v v v • H • v v v) • (1 × v v v) # order degree Formula 5 1 1 v v v 6 1 1 1 × v v v 7 1 3 v v v • H • v v v 8 1 3 v v v • H • (1 × v v v) 9 2 0 1 10 2 2 H • v v v 11 2 2 H • (1 × v v v) 12 3 1 H 13 3 1 1 × H Table 6. A minimal integrity basis for Cov(S 3 (R 2 ), O (2))
# order degree Formula

1 0 2 v v v • v v v 2 0 2 H . . . H 3 0 4 (v v v • H • v v v) • v v v # order degree Formula 4 1 1 v v v 5 1 3 v v v • H • v v v 6 2 0 1 7 2 2 H • v v v 8 3 1 H
Note that in the present case, there are pseudo-scalars which reduce to additional H 0 (R 2 ) components when restricted to SO(2): ξ 1 = λ 4 , ξ 2 = λ 5 and ξ 3 = λ 6 .

Theorem 8.5. A minimal integrity basis for Cov(T 4 (R 2 ), SO(2)) consists in the 62 covariants (43 invariants) of Table 7 and Table 8. A minimal integrity basis for Cov(T 4 (R 2 ), O(2)) consists in the 115 covariants (78 invariants) of Table 7 and Table 9.

Remark 8.6. In the case of O(2), all products Im(m p ) Im(m q ) disappear after cleaning but the products ξ i Im(m l ) remain, of course, and are listed in table Table 9. Photoelasticity tensor. The 2D photoelasticity tensor Π Π Π [START_REF] Coker | A Treatise on Photoelasticity[END_REF], like the 2D Eshelby tensor, has the following index symmetry Π ijkl = Π jikl = Π ijlk . The corresponding tensor space, noted Gel in [START_REF] Forte | Symmetry classes and harmonic decomposition for photoelasticity tensors[END_REF], has the following harmonic decomposition under SO(2) and

3H 0 (R 2 ) ⊕ 2H 2 (R 2 ) ⊕ H 4 (R 2 ),
H -1 (R 2 ) ⊕ 2H 0 (R 2 ) ⊕ 2H 2 (R 2 ) ⊕ H 4 (R 2 ),
under O(2). It corresponds to the subspace of T 4 (R 2 ), where

λ 1 = λ, λ 2 = µ, ξ 1 = ξ, λ 3 = ξ 2 = ξ 3 = 0, z 2c = z 2d = 0.
In the following corollary, we provide for it a minimal integrity basis of its covariant algebra and correct, by the way, an error in [START_REF] Ming | An irreducible polynomial functional basis of two-dimensional Eshelby tensors[END_REF]. Indeed, the O(2)-integrity basis of its invariant algebra provided there, is of cardinal 10, and omits all the irreducible invariants of Table 12.

Corollary 8.7. A minimal integrity basis for Cov(Gel, SO(2))), where we have set Remark 8.8. In [START_REF] Ming | An irreducible polynomial functional basis of two-dimensional Eshelby tensors[END_REF], the authors have forgotten the 4 products ξ i Im m l of Table 12 which cannot be removed from any minimal basis of the O(2)-invariant algebra, by theorem 5.5.

Π Π Π = (ξ, λ, µ, h 1 , h 2 , H), with ξ ∈ H -1 (R 2 ), λ, µ ∈ H 0 (R 2 ), h 1 , h 2 ∈ H 2 (R 2 ), H ∈ H 4 (R 2 ),
h 1 : (1 × h 2 ) 18 0 3 Im[z 2 2a z 4 ] h 1 : H : (1 × h 1 ) 19 0 3 Im[z 2 2b z 4 ] h 2 : H : (1 × h 2 ) 20 0 3 Im[z 2a z 2b z 4 ] h 1 : H : (1 × h 2 )
# order degree Formula Formula

21 2 1 Im[z 2a z 2 ] (1 × h 1 ) 22 2 1 Im[z 2b z 2 ] (1 × h 2 ) 23 2 2 Im[z 2a z 4 z 2 ] H : (1 × h 1 ) 24 2 2 Im[z 2b z 4 z 2 ] H : (1 × h 2 ) 25 4 1 Im[z 4 z 4 ] (1 × H)
Table 12. Isotropic products of hemitropic covariants of the photoelasticity tensor # order degree Formula Formula

16 0 2 ξ 2 ξ 2 17 0 3 ξ Im[z 2a z 2b ] ξ h 1 : (1 × h 2 ) 18 0 4 ξ Im[z 2 2a z 4 ] ξ h 1 : H : (1 × h 1 ) 19 0 4 ξ Im[z 2 2b z 4 ] ξ h 2 : H : (1 × h 2 ) 20 0 4 ξ Im[z 2a z 2b z 4 ] ξ h 1 : H : (1 × h 2 ) # order degree Formula Formula 21 2 2 ξ Im(z 2a z 2 ) ξ 1 × h 1 22 2 2 ξ Im(z 2b z 2 ) ξ 1 × h 2 23 2 3 ξ Im[z 2a z 4 z 2 ] ξ H : (1 × h 1 ) 24 2 3 ξ Im[z 2b z 4 z 2 ] ξ H : (1 × h 2 )
2H 0 (R 2 ) ⊕ H 2 (R 2 ) ⊕ H 4 (R 2 ),
both for O(2) and SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF]. It corresponds to the subspace of T 4 (R 2 ), where

λ 3 = ξ 1 = ξ 2 = ξ 3 = 0, z 2b = z 2c = z 2d = 0,
and to the subspace of Gel, where

ξ = 0, h 1 = h, h 2 = 0.
The following corollary completes the already known integrity basis of the invariant algebra of Ela (see examples 5.4, 5.10 and 7.5) into a minimal integrity basis of its covariant algebra.

Corollary 8.9. A minimal integrity basis for Cov(Ela, SO(2))), where we have set

C = (λ, µ, h, H), with λ, µ ∈ H 0 (R 2 ), h ∈ H 2 (R 2 ), H ∈ H 4 (R 2 )
, consists in the 13 covariants (6 invariants) of Table 13 (A) and Table 13 (B). A minimal integrity basis for Cov(Ela, O(2)) consists in the 9 covariants (5 invariants) of Table 13 (A).

8.3.

Viscoelasticity law and Hill elasto-plasticity. In linear viscoelasticity, the application of a periodic strain tensor at frequency f , seen as the imaginary part of ǫ ǫ ǫ = ǫ ǫ ǫ a exp(2iπf t), generates a periodic stress tensor which is the imaginary part of σ σ σ = σ σ σ a exp(i(2πf t + ϕ)); ǫ ǫ ǫ a and σ σ σ a are the assumed constant strain and stress amplitude (symmetric) tensors and ϕ is the phase shift. A frequency dependent anisotropic viscoelasticity behaviour can then be formulated as where

(8.1) σ σ σ = C * : ǫ ǫ ǫ,
C * = C * (f ) = C 1 + i C 2 (with C 1 , C 2 ∈ Ela)
is the complex viscoelasticity tensor. For the purpose we are concerned in, we will however still consider this representation as a real representation of either SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] or O( 2) and represent it as the (de-complexified) vector space V = Ela ⊕ Ela.

On the other hand, Hill elasto-plasticity constitutive equations [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF] can be summarized into the linear elasticity law σ σ σ = C : (ǫ ǫ ǫǫ ǫ ǫ p ), where C ∈ Ela is the elasticity tensor, ǫ ǫ ǫ p ∈ S 2 (R 2 ) is the plastic strain tensor, and into the (plasticity) yield criterion f = σ σ σ :

P H : σ σ σ -R(p) ≤ 0,
where P H ∈ Ela is the Hill fourth-order tensor and R is the hardening function (p being the so-called accumulated plastic strain). The evolution laws are obtained by generalized normality (see [START_REF] Lemaitre | Mécanique des matériaux solides. Dunod, english translation 1990 'Mechanics of Solid Materials[END_REF]). Hill elasto-plasticity law is also represented by a pair (C 1 , C 2 ) of tensors of the elasticity type, but this time, with C 1 = C, and C 2 = P H . The harmonic decomposition of V = Ela ⊕ Ela, which is the same for SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] and O(2), is

4H 0 (R 2 ) ⊕ 2H 2 (R 2 ) ⊕ 2H 4 (R 2 ),
and we will write

(C 1 , C 2 ) = (λ 1 , λ 2 , µ 1 , µ 2 , z 2a , z 2b , z 4a , z 4b ) = (λ 1 , λ 2 , µ 1 , µ 2 , h 1 , h 2 , H 1 , H 2 ).
Theorem 8.10. A minimal integrity basis for Cov(Ela ⊕ Ela, SO(2)) consists in the 41 covariants (24 invariants) of Table 14 and Table 15. A minimal integrity basis for Cov(Ela⊕Ela, O(2)) consists in the 28 covariants (17 invariants) of Table 14 and Table 16.

Remark 8.11. Note that, in the minimal covariant basis for Cov(Ela ⊕ Ela, O( 2)), it remains 3 products Im(m p ) Im(m q ) (see Table 16), which cannot be eliminated after cleaning.

8.4. Piezoelectricity law. The linear piezoelectricity law [START_REF] Royer | Elastic waves in solids I: Free and guided propagation[END_REF] is a linear relation between the (symmetric) second-order strain tensor ǫ ǫ ǫ, the (symmetric) second-order stress tensor σ σ σ, the electric field E E E, and the electric displacement D D D. It writes

σ σ σ = C : ǫ ǫ ǫ -E E E • P, D D D = P : σ σ σ + ε ε ε σ 0 • E E E, σ ij = C ijkl ǫ kl -P kij E k , D i = P ikl σ kl + ε σ 0ik E k ,
where C ∈ Ela is the elasticity fourth-order tensor (C ijkl = C jikl = C klij ), P ∈ Piez is the piezoelectricity third-order tensor (P kij = P kji ), and ε ε ε σ 0 is the (symmetric) second-order dielectric permittivity tensor (which should not be confused with the Levi-Civita tensor ε ε ε or the strain tensor ǫ ǫ ǫ). The harmonic decomposition of the constitutive tensor (C, P, ε ε ε σ 0 ), with respect to either SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] or O(2) writes as

3H 0 (R 2 ) ⊕ 2H 1 (R 2 ) ⊕ 2H 2 (R 2 ) ⊕ H 3 (R 2 ) ⊕ H 4 (R 2 ), Table 14. Isotropic covariants of V = Ela ⊕ Ela # order degree Formula Formula 1 1 λ 1 λ 1 2 1 µ 1 µ 1 3 1 λ 2 λ 2 4 1 µ 2 µ 2 5 2 z 2a z 2a h 1 : h 1 6 2 z 2b z 2b h 2 : h 2 7 2 z 4a z 4a H 1 :: H 1 8 2 z 4b z 4b H 2 :: H 2 9 2 Re[z 2a z 2b ] h 1 : h 2 10 2 Re[z 4a z 4b ] H 1 :: H 2 11 3 Re[z 2 2a z 4a ] h 1 : H 1 : h 1 12 3 Re[z 2 2b z 4a ] h 2 : H 1 : h 2 13 3 Re[z 2 2a z 4b ] h 1 : H 2 : h 1 14 3 Re[z 2 2b z 4b ] h 2 : H 2 : h 2 15 3 Re[z 2a z 2b z 4a ] h 1 : H 1 : h 2 16 3 Re[z 2a z 2b z 4b ] h 1 : H 2 : h 2 # order degree Formula Formula 17 2 0 zz 1 18 2 1 Re[z 2a z 2 ] h 1 19 2 1 Re[z 2b z 2 ] h 2 20 2 2 Re[z 2a z 4a z 2 ] H 1 : h 1 21 2 2 Re[z 2a z 4b z 2 ] H 2 : h 1 22 2 2 Re[z 2b z 4a z 2 ] H 1 : h 2 23 2 2 Re[z 2b z 4b z 2 ] H 2 : h 2 24 4 1 Re[z 4a z 4 ] H 1 25 4 1 Re[z 4b z 4 ] H 2 Table 15. Hemitropic covariants of V = Ela ⊕ Ela # order degree Formula Formula 26 0 2 Im[z 2a z 2b ] h 1 : (1 × h 2 ) 27 0 2 Im[z 4a z 4b ] H 1 : (1 × H 2 ) 28 0 3 Im[z 2 2a z 4a ] h 1 : H 1 : (1 × h 1 ) 29 0 3 Im[z 2 2b z 4a ] h 2 : H 1 : (1 × h 2 ) 30 0 3 Im[z 2 2a z 4b ] h 1 : H 2 : (1 × h 1 ) 31 0 3 Im[z 2 2b z 4b ] h 2 : H 2 : (1 × h 2 ) 32 0 3 Im[z 2a z 2b z 4a ] h 1 : H 1 : (1 × h 2 ) 33 0 3 Im[z 2a z 2b z 4b ] h 1 : H 2 : (1 × h 2 ) # order degree Formula Formula 34 2 1 Im[z 2a z 2 ] 1 × h 1 35 2 1 Im[z 2b z 2 ] 1 × h 2 36 2 2 Im[z 2a z 4a z 2 ] H 1 : (1 × h 1 ) 37 2 2 Im[z 2a z 4b z 2 ] H 2 : (1 × h 1 ) 38 2 2 Im[z 2b z 4a z 2 ] H 1 : (1 × h 2 ) 39 2 2 Im[z 2b z 4b z 2 ] H 2 : (1 × h 2 ) 40 4 1 Im[z 4a z 4 ] 1 × H 1 41 4 1 Im[z 4b z 4 ] 1 × H 2
Table 16. Isotropic products of hemitropic covariants of V = Ela ⊕ Ela # order degree Formula Formula

26 0 4 Im[z 2a z 2b ] Im[z 4a z 4b ] (h 1 : (1 × h 2 ))(H 1 :: (1 × H 2 )) 27 2 3 Im[z 4a z 4b ] Im[z 2a z 2 ] (H 1 :: (1 × H 2 )) (1 × h 1 ) 28 2 3 Im[z 4a z 4b ] Im[z 2b z 2 ] (H 1 :: (1 × H 2 )) (1 × h 2 )
and we will write (C, P, ε ε ε σ 0 ) = (λ 1 , λ 2 , λ 3 , z 1a , z 1b , z 2a , z 2b , z 3 , z 4 ). Theorem 8.12. A minimal integrity basis for the SO(2)-covariant algebra for the triplet (C, P, ε ε ε σ 0 ) consists in the 206 covariants (121 invariants) of Table 17 and Table 18. A minimal integrity basis for the O(2)-covariant algebra for the triplet (C, P, ε ε ε σ 0 ) consists in the 123 covariants (71 invariants) of Table 17 and Table 19. Remark 8.13. In this case also, it remains many products Im(m p ) Im(m q ) (see Table 19) which cannot be eliminated from the O(2)-integrity basis .

Table 17. Isotropic covariants for the triplet (C, P, ε ε ε σ 0 )

# order degree Formula

1 0 1 λ 1 2 0 1 λ 2 3 0 1 λ 3 4 0 2 z 1a z 1a 5 0 2 z 1b z 1b 6 0 2 z 2a z 2a 7 0 2 z 2b z 2b 8 0 2 z 3 z 3 9 0 2 z 4 z 4 10 0 2 Re[z 1a z 1b ] 11 0 2 Re[z 2a z 2b ] 12 0 3 Re[z 2 1a z 2a ] 13 0 3 Re[z 2 1a z 2b ] 14 0 3 Re[z 2 1b z 2a ] 15 0 3 Re[z 2 1b z 2b ] 16 0 3 Re[z 2 2a z 4 ] 17 0 3 Re[z 2 2b z 4 ] 18 0 3 Re[z 1a z 1b z 2a ] 19 0 3 Re[z 1a z 1b z 2b ] 20 0 3 Re[z 1a z 2a z 3 ] 21 0 3 Re[z 1b z 2a z 3 ] 22 0 3 Re[z 1b z 2b z 3 ] 23 0 3 Re[z 1a z 2b z 3 ] 24 0 3 Re[z 2a z 2b z 4 ] 25 0 3 Re[z 1a z 3 z 4 ] 26 0 3 Re[z 1b z 3 z 4 ] 27 0 4 Re[z 3 1a z 3 ] 28 0 4 Re[z 2 1a z 1b z 3 ] 29 0 4 Re[z 1a z 2 1b z 3 ] 30 0 4 Re[z 3 1b z 3 ] 31 0 4 Re[z 1a z 2 2a z 3 ] 32 0 4 Re[z 1b z 2 2a z 3 ] 33 0 4 Re[z 1a z 2a z 2b z 3 ] 34 0 4 Re[z 1b z 2a z 2b z 3 ] 35 0 4 Re[z 1a z 2 2b z 3 ] 36 0 4 Re[z 1b z 2 2b z 3 ] 37 0 4 Re[z 2 1a z 2a z 4 ] 38 0 4 Re[z 1a z 1b z 2a z 4 ] 39 0 4 Re[z 2 1b z 2a z 4 ] 40 0 4 Re[z 2 1a z 2b z 4 ] 41 0 4 Re[z 1a z 1b z 2b z 4 ] 42 0 4 Re[z 2 1b z 2b z 4 ] 43 0 4 Re[z 1a z 2a z 3 z 4 ] 44 0 4 Re[z 1b z 2a z 3 z 4 ] 45 0 4 Re[z 1a z 2b z 3 z 4 ] 46 0 4 Re[z 1b z 2b z 3 z 4 ] 47 0 4 Re[z 2a z 2 3 z 4 ] 48 0 4 Re[z 2b z 2 3 z 4 ] 49 0 5 Re[z 3 2a z 2 3 ] 50 0 5 Re[z 2 2a z 2b z 2 3 ] 51 0 5 Re[z 2a z 2 2b z 2 3 ] 52 0 5 Re[z 3 2b z 2 3 ] 53 0 5 Re[z 4 1a z 4 ] 54 0 5 Re[z 3 1a z 1b z 4 ] 55 0 5 Re[z 2 1a z 2 1b z 4 ] # order degree Formula 56 0 5 Re[z 1a z 3 1b z 4 ] 57 0 5 Re[z 4 1b z 4 ] 58 0 5 Re[z 2a z 2 3 z 2 4 ] 59 0 5 Re[z 2b z 2 3 z 2 4 ] 60 0 5 Re[z 2 1a z 2 3 z 4 ] 61 0 5 Re[z 1a z 1b z 2 3 z 4 ] 62 0 5 Re[z 2 1b z 2 3 z 4 ] 63 0 6 Re[z 1a z 3 3 z 2 4 ] 64 0 6 Re[z 1b z 3 3 z 2 4 ] 65 0 7 Re[z 4 3 z 3 4 ] 66 1 1 Re[z 1a z] 67 1 1 Re[z 1b z] 68 1 2 Re[z 1a z 2a z] 69 1 2 Re[z 1b z 2a z] 70 1 2 Re[z 1a z 2b z] 71 1 2 Re[z 1b z 2b z] 72 1 2 Re[z 2a z 3 z] 73 1 2 Re[z 2b z 3 z] 74 1 2 Re[z 3 z 4 z] 75 1 3 Re[z 2 1a z 3 z] 76 1 3 Re[z 2 2b z 3 z] 77 1 3 Re[z 2 1b z 3 z] 78 1 3 Re[z 2 2a z 3 z] 79 1 3 Re[z 2a z 2b z 3 z] 80 1 3 Re[z 1a z 1b z 3 z] 81 1 3 Re[z 2a z 3 z 4 z] 82 1 3 Re[z 2b z 3 z 4 z] 83 1 3 Re[z 1a z 2a z 4 z] 84 1 3 Re[z 1b z 2a z 4 z] 85 1 3 Re[z 1a z 2b z 4 z] 86 1 3 Re[z 1b z 2b z 4 z] 87 1 4 Re[z 3 1a z 4 z] 88 1 4 Re[z 3 1b z 4 z] 89 1 4 Re[z 1a z 2 3 z 4 z] 90 1 4 Re[z 1b z 2 3 z 4 z] 91 1 4 Re[z 2 1a z 1b z 4 z] 92 1 4 Re[z 1a z 2 1b z 4 z] 93 1 5 Re[z 3 3 z 2 4 z] 94 2 0 zz 95 2 1 Re[z 2a z 2 ] 96 2 1 Re[z 2b z 2 ] 97 2 2 Re[z 1a z 3 z 2 ] 98 2 2 Re[z 1b z 3 z 2 ] 99 2 2 Re[z 2a z 4 z 2 ] 100 2 2 Re[z 2b z 4 z 2 ] 101 2 3 Re[z 2 1a z 4 z 2 ] 102 2 3 Re[z 2 1b z 4 z 2 ] 103 2 3 Re[z 2 3 z 4 z 2 ] 104 2 3 Re[z 1a z 1b z 4 z 2 ] 105 3 1 Re[z 3 z 3 ] 106 3 2 Re[z 1a z 4 z 3 ] 107 3 2 Re[z 1b z 4 z 3 ] 108 4 1 Re[z 4 z 4 ]
Table 18. Hemitropic covariants for the triplet (C, P, ε ε ε σ 0 ) 8.5. Twelfth-order totally symmetric tensor. Totally symmetric tensors are encountered for the intrinsic description of directional data [START_REF] Kanatani | Distribution of directional data and fabric tensors[END_REF]. They may represent the directional density of spatial contacts and grains orientations within granular materials [START_REF] Oda | Fabric tensor for discontinuous geological materials[END_REF], the directional crack density [START_REF] Kanatani | Distribution of directional data and fabric tensors[END_REF][START_REF] Onat | Effective properties of elastic materials that contain penny shaped voids[END_REF][START_REF] Lubarda | Damage tensors and the crack density distribution[END_REF][START_REF] Tikhomirov | On three-dimensional microcrack density distribution[END_REF], or the directional description of microstructure degradation by rafting in single crystal superalloys at high temperature [START_REF] Caccuri | Tensorial nature of γ′-rafting evolution in nickel-based single crystal superalloys[END_REF]. The harmonic decomposition of S 12 (R 2 ) is the same under SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] or O(2) and writes

# order degree Formula 109 0 2 Im[z 1a z 1b ] 110 0 2 Im[z 2a z 2b ] 111 0 3 Im[z 2 1a z 2a ] 112 0 3 Im[z 2 1a z 2b ] 113 0 3 Im[z 2 1b z 2a ] 114 0 3 Im[z 2 1b z 2b ] 115 0 3 Im[z 2 2a z 4 ] 116 0 3 Im[z 2 2b z 4 ] 117 0 3 Im[z 1a z 1b z 2a ] 118 0 3 Im[z 1a z 1b z 2b ] 119 0 3 Im[z 1a z 2a z 3 ] 120 0 3 Im[z 1b z 2a z 3 ] 121 0 3 Im[z 1b z 2b z 3 ] 122 0 3 Im[z 1a z 2b z 3 ] 123 0 3 Im[z 2a z 2b z 4 ] 124 0 3 Im[z 1a z 3 z 4 ] 125 0 3 Im[z 1b z 3 z 4 ] 126 0 4 Im[z 3 1a z 3 ] 127 0 4 Im[z 2 1a z 1b z 3 ] 128 0 4 Im[z 1a z 2 1b z 3 ] 129 0 4 Im[z 3 1b z 3 ] 130 0 4 Im[z 1a z 2 2a z 3 ] 131 0 4 Im[z 1b z 2 2a z 3 ] 132 0 4 Im[z 1a z 2a z 2b z 3 ] 133 0 4 Im[z 1b z 2a z 2b z 3 ] 134 0 4 Im[z 1a z 2 2b z 3 ] 135 0 4 Im[z 1b z 2 2b z 3 ] 136 0 4 Im[z 2 1a z 2a z 4 ] 137 0 4 Im[z 1a z 1b z 2a z 4 ] 138 0 4 Im[z 2 1b z 2a z 4 ] 139 0 4 Im[z 2 1a z 2b z 4 ] 140 0 4 Im[z 1a z 1b z 2b z 4 ] 141 0 4 Im[z 2 1b z 2b z 4 ] 142 0 4 Im[z 1a z 2a z 3 z 4 ] 143 0 4 Im[z 1b z 2a z 3 z 4 ] 144 0 4 Im[z 1a z 2b z 3 z 4 ] 145 0 4 Im[z 1b z 2b z 3 z 4 ] 146 0 4 Im[z 2a z 2 3 z 4 ] 147 0 4 Im[z 2b z 2 3 z 4 ] 148 0 5 Im[z 3 2a z 2 3 ] 149 0 5 Im[z 2 2a z 2b z 2 3 ] 150 0 5 Im[z 2a z 2 2b z 2 3 ] 151 0 5 Im[z 3 2b z 2 3 ] 152 0 5 Im[z 4 1a z 4 ] 153 0 5 Im[z 3 1a z 1b z 4 ] 154 0 5 Im[z 2 1a z 2 1b z 4 ] 155 0 5 Im[z 1a z 3 1b z 4 ] 156 0 5 Im[z 4 1b z 4 ] 157 0 5 Im[z 2a z 2 3 z 2 4 ] 158 0 5 Im[z 2b z 2 3 z 2 4 ] # order degree Formula 159 0 5 Im[z 2 1a z 2 3 z 4 ] 160 0 5 Im[z 1a z 1b z 2 3 z 4 ] 161 0 5 Im[z 2 1b z 2 3 z 4 ] 162 0 6 Im[z 1a z 3 3 z 2 4 ] 163 0 6 Im[z 1b z 3 3 z 2 4 ] 164 0 7 Im[z 4 3 z 3 4 ] 165 1 1 Im[z 1a z] 166 1 1 Im[z 1b z] 167 1 2 Im[z 1a z 2a z] 168 1 2 Im[z 1b z 2a z] 169 1 2 Im[z 1a z 2b z] 170 1 2 Im[z 1b z 2b z] 171 1 2 Im[z 2a z 3 z] 172 1 2 Im[z 2b z 3 z] 173 1 2 Im[z 3 z 4 z] 174 1 3 Im[z 2 1a z 3 z] 175 1 3 Im[z 2 2b z 3 z] 176 1 3 Im[z 2 1b z 3 z] 177 1 3 Im[z 2 2a z 3 z] 178 1 3 Im[z 2a z 2b z 3 z] 179 1 3 Im[z 1a z 1b z 3 z] 180 1 3 Im[z 2a z 3 z 4 z] 181 1 3 Im[z 2b z 3 z 4 z] 182 1 3 Im[z 1a z 2a z 4 z] 183 1 3 Im[z 1b z 2a z 4 z] 184 1 3 Im[z 1a z 2b z 4 z] 185 1 3 Im[z 1b z 2b z 4 z] 186 1 4 Im[z 3 1a z 4 z] 187 1 4 Im[z 3 1b z 4 z] 188 1 4 Im[z 1a z 2 3 z 4 z] 189 1 4 Im[z 1b z 2 3 z 4 z] 190 1 4 Im[z 2 1a z 1b z 4 z] 191 1 4 Im[z 1a z 2 1b z 4 z] 192 1 5 Im[z 3 3 z 2 4 z] 193 2 1 Im[z 2a z 2 ] 194 2 1 Im[z 2b z 2 ] 195 2 2 Im[z 1a z 3 z 2 ] 196 2 2 Im[z 1b z 3 z 2 ] 197 2 2 Im[z 2a z 4 z 2 ] 198 2 2 Im[z 2b z 4 z 2 ] 199 2 3 Im[z 2 1a z 4 z 2 ] 200 2 3 Im[z 2 1b z 4 z 2 ] 201 2 3 Im[z 2 3 z 4 z 2 ] 202 2 3 Im[z 1a z 1b z 4 z 2 ] 203 3 1 Im[z 3 z 3 ] 204 3 2 Im[z 1a z 4 z 3 ] 205 3 2 Im[z 1b z 4 z 3 ] 206 4 1 Im[z 4 z 4 ]
H 0 (R 2 ) ⊕ H 2 (R 2 ) ⊕ H 4 (R 2 ) ⊕ H 6 (R 2 ) ⊕ H 8 (R 2 ) ⊕ H 10 (R 2 ) ⊕ H 12 (R 2 ),
and we will write S = (λ, z 2 , z 4 , z 6 , z 8 , z 10 , z 12 ), where λ ∈ R and z k ∈ C. Due to the large number of two-dimensional harmonic components, only minimal integrity bases for its invariant algebra are detailed (and, as for piezoelectricity law, translations into tensorial expressions will not be provided).

Theorem 8.14. A minimal integrity basis for Inv(S 12 (R 2 ), SO(2)) consists in the 211 invariants of Table 20

Conclusion

In this paper, we have formulated, with full details, a method to compute a minimal integrity basis for the invariant algebra of any 2D constitutive tensor (and more generally of any linear representation of the orthogonal groups SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] or O(2)). These results are formalized as theorem 5.3 and theorem 5.5. In the second case, a minimal integrity basis is obtained after applying a "cleaning algorithm" which is explained in section 6. Besides, several reduction lemmas have been proven in section 5, which allow to reduce a priori the complexity of the computations required to obtain minimality.

Meanwhile, a new paradigm has also been introduced: the concept of polynomial covariant which extends the idea of a polynomial invariant. This notion is not really new in mathematics, since it goes back to the early ages of Classical Invariant Theory in the nineteenth century. However, it was not clear how to formulate this concept in the framework of tensor spaces rather than binary forms, for which it was introduced first. This task is now achieved and the covariant algebra of a representation V of G (where G = O(2) or G = SO(2)), is defined as

Cov(V, G) := Inv(V ⊕ R 2 , G).
In other words, it is defined as the invariant algebra of the considered vector space V (and thus, in particular, for any constitutive tensor), to which is added a vector space R 2 (i.e. a vector x x x = (x, y)). This concept, as exotic as it may sound first for a non specialist, has proven to be much more useful than the invariant algebra itself to solve, for instance, such a problem as the characterization of symmetry classes of the elasticity tensor [START_REF] Olive | Characterization of the symmetry class of an elasticity tensor using polynomial covariants[END_REF] in a simple manner (compared to invariant characterization of only its fourth-order harmonic part in [START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF]). It is implicit in the work of [START_REF] Boehler | On the polynomial invariants of the elasticity tensor[END_REF] and more generally in the theory of tensorial representations [START_REF] Boehler | Application of tensor functions in solid mechanics[END_REF].

Therefore, we have computed minimal integrity bases for covariant algebras rather than invariant algebras, and this for an exhaustive list of constitutive tensors and laws (a minimal basis for the invariant algebra is however immediately deduced from a minimal basis of the covariant algebra but the converse is, of course, not true). The proposed algorithm has proven to be very effective and this is illustrated by the fact that we have been able to compute a minimal integrity basis for the covariant algebra of the following constitutive tensors and laws: all third order tensors (including the piezoelectricty tensor), all fourth-order tensors (including the elasticity, the Eshelby and the photoelasticity tensors), the linear viscoelasticity law, the Hill elasto-plasticity constitutive equations, the linear (coupled) piezoelectricity law, and (by theorem 5.11) any even order symmetric tensors up to order 12.

Our method relies first on an explicit harmonic decomposition of the given tensor space. The means to achieve this first task are explained with full details in Appendix A and Appendix B. This allows us then to parameterize a tensor by an n-uple of complex numbers (z 1 , . . . , z r ) together with some real parameters λ k (some isotropic invariants) and ξ i (some hemitropic invariants) but which do not enter explicitly in the computing process. A finite integrity basis is then obtained by solving a Diophantine equation which must be satisfied by the exponents of the monomials

m = z α 1 1 • • • z αr r z β 1 1 • • • z βr r .
Integrity bases are thus formulated first using these complex variables (together with the λ k and ξ i ). Moreover, a rigourous and quite exhaustive process has been achieved in section 7 to translate all these expressions into tensorial ones, since those are familiar to the mechanical community. Minimal integrity bases for most common constitutive tensors and laws have been expressed this way.

an explicit harmonic decomposition, other approaches are discussed, for instance, in [109]. The procedure described here follows moreover the same lines as Verchery's original construction for the elasticity tensor [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF], and applied later to the piezoelectricity tensor by Vannucci [START_REF] Vannucci | The polar analysis of a third order piezoelectricity-like plane tensor[END_REF]. The methodology consists in the following steps.

(1) Consider first the linearized action of SO(2) on V, or more precisely, the induced representation ρ ′ of the Lie algebra so(2) of SO(2) on V. The Lie algebra, so(2), is one-dimensional and spanned by

u := d dθ θ=0 r θ = 0 -1 1 0 .
This action is given by

ρ ′ (u)v v v = d dθ θ=0 ρ(r θ )v v v.
(2) Using an SO(2)-invariant inner-product •, • on V, we have

ρ(g)v v v 1 , ρ(g)v v v 2 = v v v 1 , v v v 2 , for all v v v 1 , v v v 2 ∈
V and for all g ∈ SO(2) and thus

ρ ′ (u)v v v 1 , v v v 2 = -v v v 1 , ρ ′ (u)v v v 2 .
In other words, ρ ′ (u) is a skew-symmetric linear endomorphism of V relatively to this invariant inner product.

Remark A.1. An O(2)-invariant inner-product on T n (R 2 ) is given, using (2.2), by

T 1 , T 2 := T 1 (n) • T 2
and can be extended into an hermitian product on the complexification (T n )

C := T n ⊕ iT n of T n (R 2 ), by T 1 , T 2 := T 1 (n) • T 2 .
The matrix representation [ρ ′ (u)] of ρ ′ (u) in the basis (e e e i 1 ⊗ . . . ⊗ e e e in ) of T n (R

A.2) (ρ ′ (u)T) i 1 ...in = n k=1 u i k j k T i 1 ...i k-1 j k i k+1 ...in .
In practice, we usually consider representations which are stable subspaces V of T n (R 2 ) rather than T n (R 2 ) itself. Typically, V is a subspace of T n (R 2 ) defined by some index symmetries. An orthonormal basis of V consists then of linear combinations of e e e i 1 ⊗ . . . ⊗ e e e in (see examples A.4, A.5 and A.6, for instance).

(3) Since ρ ′ (u) is skew-symmetric, its eigenvalues are pure imaginary complex numbers. Moreover, since we know a priori that ρ decomposes into harmonic factors, we can conclude that these eigenvalues write i n, where n ∈ Z are relative integers. Each vanishing eigenvalue corresponds to a factor H 0 (R 2 ), while each pair of eigenvalues (i n, -i n) (n ≥ 1) corresponds to a factor H n (R 2 ) (with possible multiplicity). Therefore, the de-complexification of an orthonormal basis (with respect to the hermitian product on the complexification space V C := V ⊕ iV) of eigenvectors for ρ ′ (u) provides an explicit harmonic decomposition relatively to SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF]. Let denote by U k the (real) unit eigenvectors associated with the corresponding vanishing eigenvalues, by W l the (complex) unit eigenvectors associated with the eigenvalues i n l (when n l ≥ 1) and by W l , their complex conjugates, associated with the eigenvalue -i n l .

Remark A.2. Note that each real unit eigenvector U k is defined up to a sign whereas each complex unit eigenvector W l is defined up to a phase or in other words up to a complex number e iψ l . This ambiguity is well-known in Quantum Mechanics and discussed, for instance, in [2, section II.B.1].

(4) When a representation of O( 2) is involved, one starts by calculating an harmonic decomposition relatively to the subgroup SO(2) and then checks, for each vanishing eigenvalue, whether the corresponding eigenvector U k satisfies ρ(σ)U k = U k or ρ(σ)U k = -U k . In the first case, such an eigenvector will still be denoted by U k but in the second case, it will be changed to V i . Note now that

ρ(r θ )U k = U k , ρ(r θ )V i = V i , ρ(r θ )W l = e in l θ W l ,
independently of the choice of the eigenvectors U k , W l . However, the problem is slightly more subtle for σ. Indeed, the C-linear extension of the representation ρ satisfies

ρ(σ)U k = U k , ρ(σ)V i = -V i , ρ(σ)W l = W l ,
only if the arbitrary phase ψ l in the choice of W l has been chosen so that

ρ(σ)W l = W l ,
which happens only for two values of ψ l (which differ by π). This must be calculated in order to produce an explicit harmonic decomposition relative to the group O(2) and fixes, by the way, the ambiguity. In the following examples, this choice has been made. Finally, one ends up with the following orthonormal decompositions of T ∈ V ⊂ T n (R 2 ). SO(2)-decomposition:

T = ν 0 k=1 λ k U k + r l=1 z l W l + z l W l , O (2)
-decomposition:

T = m 0 k=1 λ k U k + m -1 i=1 ξ i V i + r l=1 z l W l + z l W l , where λ k = T, U k ∈ R, ξ i = T, V i ∈ R, z l = T, W l ∈ C. Remark A.3. Writing z l = a l + ib l , we get z l W l + z l W l = 2 (a l Re(W l ) + b l Im(W l )) .
The factor 2 here is meaningless, since the harmonic components are defined up to a scaling factor. The basis, (Re(W l ), Im(W l )) of the factor H n l (R 2 ) in the decomposition of T is orthogonal but not orthonormal, since

Re(W l ) = Im(W l ) = 1 √ 2 .
But the same is true for the basis (K

(n) 1 , K (n)
2 ) of H n (R 2 ), as defined in (7.1), it is orthogonal but not orthonormal, since

K (n) 1 = K (n) 2 = 2 n-1 2 . Example A.4. Let a ∈ V = S 2 (R 2 ) ⊂ T 2 (R 2
ρ(r θ )] =    cos 2 θ sin 2 θ -sin 2θ √ 2 sin 2 θ cos 2 θ sin 2θ √ 2 sin 2θ √ 2 -sin 2θ √ 2 cos 2θ    , [ρ ′ (u)] =   0 0 - √ 2 0 0 √ 2 √ 2 - √ 2 0   .
The eigenvalues of [ρ ′ (u)] read 0, 2i, -2i and the associated eigenvectors are respectively

[U] = (U 11 , U 22 , √ 2 U 12 ) t = ( 1 √ 2 , 1 √ 2 , 0) t , [W 2 ] = ((W 2 ) 11 , (W 2 ) 22 , √ 2 (W 2 ) 12 ) t = ( 1 2 , -1 2 , -i √ 2 ) t and [W 2 ], so that in O(2) case λ = a, U = 1 √ 2 tr a, z 2 = a, W 2 = a ′ 11 + ia 12 ,
where a ′ where (e e e 1 , e e e 2 ) is an orthonormal basis of R 2 . Using (A.2) one obtains

(A.3) (ρ ′ (u)T) ijk = u ip T pjk + u jp T ipk + u kp T ijp .

Setting

] of [ρ ′ (u)]), one gets [ρ ′ (u)T] = [ρ ′ (u)][T],
where

[ρ ′ (u)] =             0 0 0 0 0 -1 -1 -1 0 0 0 0 -1 0 1 1 0 0 0 0 -1 1 0 1 0 0 0 0 -1 1 1 0 0 1 1 1 0 0 0 0 1 0 -1 -1 0 0 0 0 1 -1 0 -1 0 0 0 0 1 -1 -1 0 0 0 0 0            
.

[ρ ′ (u)] is indeed skew-symmetric, with eigenvalues i, i, i, 3i and their four conjugates, with corresponding eigenvectors [W 1a ] = 1 2 (1, 0, 0, 1, -i, 0, 0, -i) t , [W 1b ] = 1 2 √

3 (1, 0, 2, -1, -i, 0, -2i, i) t , [W 1c ] = 1 2 √ 6 (1, 3, -1, -1, -i, -3i, i, i) t , [W 3 ] = 1 2 √ 2 (1, -1, -1, -1, i, -i, -i, -i) t and their four conjugates. This corresponds to an harmonic decomposition 3H 1 (R 2 )⊕ H 3 (R 2 ) of T 3 (R 2 ), where

z 1a = T, W 1a =
[ρ ′ (u)] =         0 0 0 0 -1 - √ 2 0 0 0 -1 0 √ 2 0 0 0 - √ 2 √ 2 1 0 1 √ 2 0 0 0 1 0 - √ 2 0 0 0 √ 2 - √ 2 -1 0 0 0        
which is indeed skew-symmetric, with eigenvalues i, i, 3i and their three conjugates with corresponding eigenvectors [W 1a ] = where β k 1 ,...,kr = 0 when at least one of the k i is odd and β k 1 ,...,kr = 1 otherwise.

Remark C.4. When the harmonic decomposition of the representation (V, ρ) of O(2) involves some components H -1 (R 2 ) and/or H 0 (R 2 ), then one can easily obtain its Hilbert series by modifying the series provided in theorem C.2, using the general formula given in theorem C.1 and observing that det(It 0 ρ 0 (r θ )) = 1t 0 , det(It 0 ρ 0 (σr θ )) = 1t 0 ,

det(I -t -1 ρ -1 (r θ )) = 1 -t -1 , det(I -t -1 ρ -1 (σr θ )) = 1 + t -1 ,
where ρ 0 and ρ -1 are defined in section 3.

Remark C.5. For a linear representation V of SO(2), which decomposes as Given two homogeneous polynomials p 1 , p 2 of respective degree n 1 and n 2 , we have then ∂ r p 2 ∂z (r-k) ∂z k . Now, if we apply this formula to two harmonic polynomials h 1 = Re(z 1 z n 1 ) and h 2 = Re(z 2 z n 2 ) with 1 ≤ r ≤ n 1 ≤ n 2 , the only non-vanishing terms in the sum correspond to k = 0 and k = r. We get thus (D.1) {h 1 , h 2 } r = 2 r-1 (zz) n 1 -r Re z 1 z 2 z n 2 -n 1 .

V ≃ ν 0 H 0 (R 2 ) ⊕ H n 1 (R 2 ) ⊕ • • • ⊕ H nr (R 2 ),
On the other hand, we have Applying this formula, first, to h 1 = Re(z 1 z n 1 ) and h 2 = Re(z 2 z n 2 ), where 1 ≤ n 1 ≤ n 2 , we get

(D.2) [h 1 , h 2 ] = (zz) n 1 -1 Im z 1 z 2 z n 2 -n 1 ,
and, then, to the Euclidean metric q := x 2 + y 2 = zz and h 1 = Re(z 1 z n 1 ), where 1 ≤ n 1 , we get (D.3) [q, h 1 ] = Im(z 1 z n 1 ) = h 1 .

Next, observe that given p harmonic polynomials h 1 , . . . , h p , where h k = Re(z k z n k ), the leading harmonic part of the product h 1 • • • h p writes (D.4)

(h 1 • • • h p ) 0 = (Re(z 1 z n 1 ) • • • Re(z p z np )) 0 = 1 2 p-1 Re(z k 1 • • • z kp z n k 1 +•••+n kp )
according to Appendix B. For instance, if 1 ≤ n 1 ≤ n 2 ≤ n 3 and n 3 ≤ n 1 + n 2 , we have Re(z 1 z n 1 ) Re(z 2 z n 2 ) Re(z 3 z n 3 ) = 1 2 2 Re(z 1 z 2 z 3 z n 1 +n 2 +n 3 ) + q n 1 Re(z 1 z 2 z 3 z -n 1 +n 2 +n 3 ) +q n 2 Re(z 1 z 2 z 3 z n 1 -n 2 +n 3 ) + q n 3 Re(z 1 z 2 z 3 z n 1 +n 2 -n 3 ) , and its leading harmonic term, of degree n 1 + n 2 + n 3 in z, z, is (Re(z 1 z n 1 ) Re(z 2 z n 2 ) Re(z 3 z n 3 )) 0 = 1 2 2 Re(z 1 z 2 z 3 z n 1 +n 2 +n 3 ). Finally, given an homogeneous harmonic polynomial h of degree n and r ≥ 0, we have △ r (q r h) = 4 r r!(n + r)! n! h, and thus, for n 1 ≤ n 2 , we get by (D.2)

(D.5) △ n 1 -1 [h 1 , h 2 ] = 4 n 1 -1 (n 1 -1)!(n 2 -1)! (n 2 -n 1 )! Im z 1 z 2 z n 2 -n 1 ,
since Im (z 1 z 2 z n 2 -n 1 ) is of degree n 2n 1 .

Proof of theorem 7.2. Let H k ∈ H n k (R 2 ), where n k ≥ 1 and set h k = φ(H k ). Then,

h k = Im(z k z n k ) = [q, h k ] = φ(1 × H k ),
by (D.3), and as 1 × H k translates as [q, h k ], according to section 2.

Proof of theorem 7.3. Let H j ∈ H n k j (R 2 ) be harmonic tensors where n k j ≥ 1, for 1 ≤ j ≤ p and set φ(H j ) = h j = Re(z j z n k j ). Then, φ (H 1 ⊙ • • • ⊙ H p ) = h 1 • • • h p and by (D.4), we get thus

φ (H 1 ⊙ • • • ⊙ H p ) ′ = (h 1 • • • h p ) 0 = 1 2 p-1 Re(z 1 • • • z p z n k 1 +•••+n kp ).
Proof of theorem 7.4. Let H j ∈ H n k j (R 2 ) be harmonic tensors where n k j ≥ 1 for 1 ≤ j ≤ p + s and set φ(H j ) = h j = Re(z j z n k j ),

Z 1 = z 1 • • • z p , Z 2 = z p+1 • • • z p+s , N 1 = n k 1 + • • • + n kp and N 2 = n k p+1 + • • • + n k p+s .
There is no loss of generality in assuming that N 1 ≤ N 2 . Then, by (D.1) with r = N 1 , we get

Re z 1 • • • z p z p+1 • • • z p+s z N 2 -N 1 = Re(Z 1 Z 2 z N 2 -N 1 ) = 1 2 N 1 -1 {H 1 , H 2 } N 1 ,
where H i := Re(Z i z N i ) for i = 1, 2. But, by (D.4), we have

H 1 = 2 p-1 (h 1 • • • h p ) 0 = 2 p-1 φ (H 1 ⊙ • • • ⊙ H p ) ′ , and H 2 = 2 s-1 (h p+1 • • • h p+s ) 0 = 2 s-1 φ (H p+1 ⊙ • • • ⊙ H p+s ) ′ . But, according to section 2 {H 1 , H 2 } N 1 = 2 p+s-2 φ (H 1 ⊙ • • • ⊙ H p ) ′ (N 1 ) • (H p+1 ⊙ • • • ⊙ H p+s ) ′ .
We get therefore

Re z 1 • • • z p z p+1 • • • z p+s z N 2 -N 1 = 2 (p+s-1-N 1 ) φ (H 1 ⊙ • • • ⊙ H p ) ′ (N 1 ) • (H p+1 ⊙ • • • ⊙ H p+s ) ′ ,
which is the first identity of theorem 7.4. Next, by (D.5), we get

Im z 1 • • • z p z p+1 • • • z p+s z N 2 -N 1 = Im(Z 1 Z 2 z N 2 -N 1 ) = (N 2 -N 1 )! 4 N 1 -1 (N 1 -1)!(N 2 -1)! △ N 1 -1 [H 1 , H 2 ] .

1 ) k e e e n-2k 1 ⊙ e e e 2k 2 1 ⊙ e e e 2k+1 2 ,

 11212

  ) and corresponds to the subspace w w w = 0 of Piez. Corollary 8.4. A minimal integrity basis for Cov(S 3 (R 2 ), SO(2)), where we have set S

25 4 2 ξ

 2 Im(z 4 z 4 ) ξ 1 × H Elasticity tensor. The 2D elasticity tensor C has the index symmetry C ijkl = C jikl = C ijlk = C klij . The corresponding tensor space Ela has the following harmonic decomposition

1 √ 2 (

 12 ) be a symmetric second order tensor and e e e 1 ⊗ e e e 1 , e e e 2 ⊗e e e 2 , e e e 1 ⊗ e e e 2 + e e e 2 ⊗ e e e 1 ), be an orthonormal basis of V. Then ρ(g)a = gag t writes [ρ(g)a] = [ρ(g)][a] with [a] = (a 11 , a 22 , √ 2 a 12 ) t and [

11 = 1 2 (a 11 -

 11211 a 22 ) is the first component of the deviatoric part a ′ = a -1 2 (tr a) 1. Example A.5. Let V = T 3 (R 2 ). Then, an orthonormal basis is given by e 111 = e e e 1 ⊗ e e e 1 ⊗ e e e 1 , e 221 = e e e 2 ⊗ e e e 2 ⊗ e e e 1 , e 122 = e e e 1 ⊗ e e e 2 ⊗ e e e 2 , e 212 = e e e 2 ⊗ e e e 1 ⊗ e e e 2 , e 222 = e e e 2 ⊗ e e e 2 ⊗ e e e 2 , e 112 = e e e 1 ⊗ e e e 1 ⊗ e e e 2 , e 211 = e e e 2 ⊗ e e e 1 ⊗ e e e 1 , e 121 = e e e 1 ⊗ e e e 2 ⊗ e e e 1 ,

where 1 ≤ 1 ( 1 - 1 ( 1 -= ( 1 -

 111111 n 1 ≤ • • • ≤ n r , the Hilbert series of Inv(V, SO(2)) writesH(t 0a , t 0b , . . . , t 1 , . . . , t r ) = 1 2πi t 0a )(1t 0b ) • • • |z|=1 r k=1 t k z n k )(1t k z -n k ) z -1 dz.Proof of theorem C.2. Observe first that det(It n ρ n (σr θ )) = 1t 2 n , ∀n ≥ 1, where the notation ρ n has been introduced in section 3. Now, from Molien-Weyl formulae (see theorem C.1) and the expression of the Haar measure on O(2) (see definition 5.7), we deduce thatH ρ (t 1 , . . . , t r )t k ρ n k (r θ )) dθ.But, for all n ≥ 1, we havedet(Itρ n (r θ )) = 1 + t 2 -2t cos(nθ) te inθ )(1te -inθ ) = (1tz n )(1tz -n ), z = e iθ, and thus I(t 1 , . . . , t r ) = 1 2πi |z|=1 r k=1

{p 1 ,∂ r p 1

 11 p 2 } r = (n 1r)!(n 2r)! n 1 !n 2 ! ∆ r αβ p 1 (x x x α )p 2 (x x x β ) x x xα=x x x β =x x x = 2 r (n 1r)!(n 2r)! n 1 !n 2 ! D αβ + D βα r p 1 (x x x α )p 2 (x x x β ) x x xα=x x x β =x x x = 2 r (n 1r)!(n 2r)! n 1 !n 2 ! ∂z k ∂z (r-k)

[p 1 ,

 1 p 2 ] = -1 n 1 n 2 (Ω αβ p 1 (x x x α )p 2 (x x x β )) x x xα=x x x β =x x x = 2i n 1 n 2 (D αβ -D βα )p 1 (x x x α )p 2 (x x x β ) x x xα=x x x β =x x x

  Definition 3.2. A representation is irreducible if there is no stable subspace under G other than {0} and V. Two linear representations (V 1 , ρ 1 ) and (V 2 , ρ 2 ) of the same group G are equivalent if there exists a linear isomorphism

	Definition 3.3.

  Example 4.8. The harmonic components of C ∈ Ela, h and H (see remark 3.6), as well as the symmetric second-order tensor d 2 (H) := H . . . H introduced in[START_REF] Boehler | On the polynomial invariants of the elasticity tensor[END_REF], are polynomial covariants of C of respective type H 2 (R 2 ), H 4 (R 2 ) and S 2 (R 2 ), for the actions of both groups SO(2) and O(2).

  n kp is a subset of {n 1 , . . . , n p }. Moreover, given any minimal integrity basis MB of Inv(V, O(2)), which consists only of multi-homogenous invariants, a minimal integrity basis of Inv(W, O(2)) is obtained by extracting, from MB, multi-homogenous invariants which depend only on the variables H n k 1 , . . . , H n kp .

Table 1 .

 1 . A minimal integrity basis for Cov(T 3 (R 2 ), O(2)) is provided by the 31 covariants (17 invariants) of Table1. Isotropic covariants of T 3 (R 2 )

	# order degree Formula	# order degree Formula
	1	2	z 1a z 1a	18	1	1	Re[z 1a z]
	2	2	z 1b z 1b	19	1	1	Re[z 1b z]
	3	2	z 1c z 1c	20	1	1	Re[z 1c z]
	4 5 6	2 2 2	z 3 z 3 Re[z 1a z 1b ] Re[z 1a z 1c ]	21 22 23	1 1 1	3 3 3	Re[z 2 1a z 3 z] Re[z 2 1b z 3 z] Re[z 2 1c z 3 z]
	7 8 9 10 11 12 13 14 15 16	2 4 4 4 4 4 4 4 4 4	Re[z 1b z 1c ] Re[z 3 1a z 3 ] Re[z 3 1b z 3 ] Re[z 3 1c z 3 ] Re[z 1a z 2 1b z 3 ] Re[z 1b z 2 1c z 3 ] Re[z 2 1b z 1c z 3 ] Re[z 2 1a z 1c z 3 ] Re[z 2 1a z 1b z 3 ] 1c z 3 ] Re[z 1a z 2	24 25 26 27 28 29 30 31	1 1 1 2 2 2 2 3	3 3 3 0 2 2 2 1	Re[z 1a z 1b z 3 z] Re[z 1a z 1c z 3 z] Re[z 1b z 1c z 3 z] zz Re[z 1a z 3 z 2 ] Re[z 1b z 3 z 2 ] Re[z 1c z 3 z 2 ] Re[z 3 z 3 ]
	17	4	Re[z 1a z 1b z 1c z 3 ]

Table 2 .

 2 Hemitropic covariants of T 3 (R 2 )

	#	order degree Formula	#	order degree Formula
	32	2	Im[z 1a z 1b ]	45	1	1	Im[z 1a
	33	2	Im[z 1a z 1c ]
	34 35 36 37 38 39 40 41 42 43 44	2 4 4 4 4 4 4 4 4 4 4	Im[z 1b z 1c ] Im[z 3 1a z 3 ] Im[z 3 1b z 3 ] Im[z 3 1c z 3 ] Im[z 1a z 2 1b z 3 ] Im[z 1b z 2 1c z 3 ] Im[z 2 1b z 1c z 3 ] Im[z 2 1a z 1c z 3 ] Im[z 2 1a z 1b z 3 ] Im[z 1a z 2 1c z 3 ] Im[z 1a z 1b z 1c z 3 ]

Table 3 .

 3 Corollary 8.3. A minimal integrity basis of Cov(Piez, SO(2)), where we have set P = (v v v, w w w, H), with v v v, w w w ∈ H 1 (R 2 ) and H ∈ H 3 (R 2 ), consists in the 30 covariants (13 invariants) of Table3. A minimal integrity basis of Cov(Piez, O(2)) consists in the 17 covariants (8 invariants) of Table 4. A minimal integrity basis for Cov(Piez, SO(2))

). By theorem 5.11, minimal integrity bases for Cov(Piez, G) (G = O(2) or G = SO(2)) are obtained by setting z 1c = 0 in theorem 8.2 (with z 1a , z 1b , z 3 defined as in example A.6). Using translation formulas of section 7, we deduce the following corollary, which completes partial results obtained by Vannucci in

[START_REF] Vannucci | The polar analysis of a third order piezoelectricity-like plane tensor[END_REF]

.

Table 7 .

 7 Isotropic covariants of T 4 (R 2 )

	#	order degree Formula	#	order degree Formula
	1	0	1	λ 1	19	0	3	Re[z 2a z 2b z 4 ]
	2	0	1	λ 2	20	0	3	Re[z 2a z 2c z 4 ]
	3	0	1	λ 3	21	0	3	Re[z 2a z 2d z 4 ]
	4	0	2	z 2a z 2a	22	0	3	Re[z 2b z 2c z 4 ]
	5	0	2	z 2b z 2b	23	0	3	Re[z 2b z 2d z 4 ]
	6	0	2	z 2c z 2c	24	0	3	Re[z 2c z 2d z 4 ]
	7 8 9 10 11 12 13 14 15 16 17 18	0 0 0 0 0 0 0 0 0 0 0 0	2 2 2 2 2 2 2 2 3 3 3 3	z 2d z 2d z 4 z 4 Re[z 2a z 2b ] Re[z 2a z 2c ] Re[z 2a z 2d ] Re[z 2b z 2c ] Re[z 2b z 2d ] Re[z 2c z 2d ] Re[z 2 2a z 4 ] Re[z 2 2b z 4 ] Re[z 2 2c z 4 ] 2d z 4 ] Re[z 2	25 26 27 28 29 30 31 32 33 34	2 2 2 2 2 2 2 2 2 4	0 1 1 1 1 2 2 2 2 1	zz Re[z 2a z 2 ] Re[z 2b z 2 ] Re[z 2c z 2 ] Re[z 2d z 2 ] Re[z 2a z 4 z 2 ] Re[z 2b z 4 z 2 ] Re[z 2c z 4 z 2 ] Re[z 2d z 4 z 2 ] Re[z 4 z 4 ]

Table 8 .

 8 Hemitropic covariants of T 4 (R 2 )

	#	order degree Formula	#	order degree Formula
	35	0	1	ξ 1 = λ 4	50	0	3	Im[z 2a z 2d z 4 ]
	36	0	1	ξ 2 = λ 5	51	0	3	Im[z 2b z 2c z 4 ]
	37	0	1	ξ 3 = λ 6	52	0	3	Im[z 2b z 2d z 4 ]
	38	0	2	Im[z 2a z 2b ]	53	0	3	Im[z 2c z 2d z 4 ]
	39 40 41 42 43 44 45 46 47 48	0 0 0 0 0 0 0 0 0 0	2 2 2 2 2 3 3 3 3 3	Im[z 2a z 2c ] Im[z 2a z 2d ] Im[z 2b z 2c ] Im[z 2b z 2d ] Im[z 2c z 2d ] Im[z 2 2a z 4 ] Im[z 2 2b z 4 ] Im[z 2 2c z 4 ] Im[z 2 2d z 4 ] Im[z 2a z 2b z 4 ]	54 55 56 57 58 59 60 61 62	2 2 2 2 2 2 2 2 4	1 1 1 1 2 2 2 2 1	Im[z 2a z 2 ] Im[z 2b z 2 ] Im[z 2c z 2 ] Im[z 2d z 2 ] Im[z 2a z 4 z 2 ] Im[z 2b z 4 z 2 ] Im[z 2c z 4 z 2 ] Im[z 2d z 4 z 2 ] Im[z 4 z 4 ]
	49	0	3	Im[z 2a z 2c z 4 ]

Table 9 .

 9 Isotropic products of hemitropic covariants of T 4 (R 2 ) (i = 1, 2, 3)

	#	order degree Formula	#	order degree Formula
	35 36 37	0 0 0	2 2 2	ξ 2 1 ξ 2 2 ξ 2 3	71, 72, 73 74, 75, 76 77, 78, 79	0 0 0	4 4 4	ξ i Im[z 2a z 2b z 4 ] ξ i Im[z 2a z 2c z 4 ] ξ i Im[z 2a z 2d z 4 ]
	38	0	2	ξ 1 ξ 2	80, 81, 82	0	4	ξ i Im[z 2b z 2c z 4 ],
	39	0	2	ξ 1 ξ 3	83, 84, 85	0	4	ξ i Im[z 2b z 2d z 4 ]
	40	0	2	ξ 2 ξ 3	86, 87, 88	0	4	ξ i Im[z 2c z 2d z 4 ]
	41, 42, 43 44, 45, 46 47, 48, 49 50, 51, 52 53, 54, 55 56, 57, 58 59, 60, 61 62, 63, 64 65, 66, 67 68, 69, 70	0 0 0 0 0 0 0 0 0 0	3 3 3 3 3 3 4 4 4 4	ξ i Im[z 2a z 2b ] ξ i Im[z 2a z 2c ] ξ i Im[z 2a z 2d ] ξ i Im[z 2b z 2c ] ξ i Im[z 2b z 2d ] ξ i Im[z 2c z 2d ] ξ i Im[z 2 2a z 4 ] ξ i Im[z 2 2b z 4 ] ξ i Im[z 2 2c z 4 ] ξ i Im[z 2 2d z 4 ]	89, 90, 91 92, 93, 94 95, 96, 97 98, 99, 100 101, 102, 103 104, 105, 106 107, 108, 109 110, 111, 112 113, 114, 115	2 2 2 2 2 2 2 2 4	2 2 2 2 3 3 3 3 2	ξ i Im(z 2a z 2 ) ξ i Im(z 2b z 2 ) ξ i Im(z 2c z 2 ) ξ i Im(z 2d z 2 ) ξ i Im[z 2a z 4 z 2 ] ξ i Im[z 2b z 4 z 2 ] ξ i Im[z 2c z 4 z 2 ] ξ i Im[z 2d z 4 z 2 ] ξ

i Im(z 4 z 4 )

Table 10 .

 10 Isotropic covariants of the photoelasticity tensor

	# order degree Formula	Formula	#	order degree Formula	Formula
	1	0	1	λ	λ	10	2	0	zz	1
	2	0	1	µ	µ	11	2	1	Re[z 2a z 2 ]	h 1
	3 4 5	0 0 0	2 2 2	z 2a z 2a z 2b z 2b z 4 z 4	h 1 : h 1 h 2 : h 2 H :: H	12 13 14	2 2 2	1 2 2	h 2 Re[z 2a z 4 z 2 ] H : h 1 Re[z 2b z 2 ] Re[z 2b z 4 z 2 ] H : h 2
	6 7 8 9	0 0 0 0	2 3 3 3	Re[z 2a z 2b ] Re[z 2 Re[z 2a z 2b z 4 ] h 1 : H : h 2 h 1 : h 2 2b z 4 ] h 2 : H : h 2 Re[z 2 2a z 4 ] h 1 : H : h 1	15	4	1	Re[z 4 z 4 ]	H

Table 11 .

 11 Hemitropic covariants of the photoelasticity tensor

	#	order degree Formula	Formula
	16	0	1	ξ	ξ
	17	0	2	Im[z 2a z 2b ]

Table 13 .

 13 Covariants of Ela

	(a) Isotropic covariants		(b) Hemitropic covariants
	# order degree Formula	#	order degree Formula
	1 2 3 4	0 0 0 0	1 1 2 2	λ µ h : h H :: H	10 11 12	0 2 2	3 1 2	h : H : (1 × h) 1 × h H : (1 × h)
	5	0	3	h : H : h	13	4	1	1 × H
	6	2	0	1
	7	2	1	h
	8	2	2	H : h
	9	4	1	H

Table 19 .

 19 Isotropic products of hemitropic covariants for the triplet (C, P, ε ε ε σ 0 )

	#	order degree Formula	#	order degree Formula
	109 110 111 112 113 114	0 0 0 0 0 0	4 5 5 5 5 5	Im[z 1a z 1b ] Im[z 2a z 2b ] Im[z 1a z 1b ] Im[z 2 2a z 4 ] Im[z 1a z 1b ] Im[z 2 2b z 4 ] Im[z 1a z 1b ] Im[z 2a z 2b z 4 ] Im[z 2a z 2b ] Im[z 1a z 3 z 4 ] Im[z 2a z 2b ] Im[z 1b z 3 z 4 ]	115 116 117 118 119 120 121	2 2 2 2 2 2 2	3 3 4 4 4 4 4	Im[z 2a z 2b ] Im[z 1a z] Im[z 2a z 2b ] Im[z 1b z] Im[z 2 2a z 4 ] Im[z 1a z] Im[z 2 2a z 4 ] Im[z 1b z] Im[z 2 2b z 4 ] Im[z 1a z] Im[z 2 2b z 4 ] Im[z 1b z] Im[z 2a z 2b z 4 ] Im[z 1a z]
					122	2	4	Im[z 2a z 2b z 4 ] Im[z 1b z]
					123	2	4	Im[z 2a z 2b ] Im[z 3 z 4 z]

Table 20 .

 20 and Table 21. A minimal integrity basis for Inv(S 12 (R 2 ), O(2)) consists in the 113 invariants of Table 20 and Table 22. Remark 8.15. Again, it remains four products Im(m p ) Im(m q ) (see Table 22) which cannot be eliminated from the O(2)-integrity basis . Isotropic invariants of S 12 (R 2 )

	#	order degree Formula
	1	0	λ
	2	0	z 2 z 2
	3	0	z 4 z 4
	4	0	z 6 z 6
	5	0	z 8 z 8
	6	0	z 10 z 10
	7	0	z 12 z 12
	8 9 10 11	0 0 0 0	Re[z 4 z 2 2 ] Re[z 8 z 2 4 ] Re[z 12 z 2 6 ] Re[z 12 z 10 z 2 ]
	12	0	Re[z 6 z 2 z 4 ]
	13	0	Re[z 8 z 2 z 6 ]
	14	0	Re[z 10 z 4 z 6 ]
	15	0	Re[z 10 z 2 z 8 ]
	16	0	Re[z 12 z 4 z 8 ]
	17 18 19 20 21 22 23 24 25 26 27 28 29	0 0 0 0 0 0 0 0 0 0 0 0 0	Re[z 6 z 3 2 ] Re[z 12 z 3 4 ] Re[z 2 6 z 10 z 2 ] Re[z 2 8 z 12 z 4 ] Re[z 8 z 2 2 z 4 ] Re[z 10 z 2 z 2 4 ] Re[z 2 4 z 2 z 6 ] Re[z 10 z 2 2 z 6 ] Re[z 2 10 z 12 z 8 ] Re[z 12 z 2 2 z 8 ] Re[z 2 6 z 4 z 8 ] Re[z 2 8 z 10 z 6 ] Re[z 4 z 8 z 10 z 2 ]
	30	0	Re[z 10 z 4 z 12 z 2 ]
	31	0	Re[z 6 z 8 z 12 z 2 ]
	32	0	Re[z 6 z 8 z 10 z 4 ]
	33	0	Re[z 10 z 6 z 12 z 4 ]
	34	0	Re[z 10 z 8 z 12 z 6 ]
	35	0	Re[z 12 z 2 z 4 z 6 ]
	36	0	Re[z 4 z 6 z 2 z 8 ]
	37

Table 21 .

 21 Hemitropic invariants of S 12 (R 2 )

	#	order degree Formula	#	order degree Formula
	110 111 112 113 114	0 0 0 0 0	3 3 3 3 3	Im[z 4 z 2 2 ] Im[z 8 z 2 4 ] Im[z 12 z 2 6 ] Im[z 12 z 10 z 2 ] Im[z 6 z 2 z 4 ]	161 162 163 164 165	0 0 0 0 0	5 5 5 5 5	Im[z 10 z 6 z Im[z 2 12 z z 6 z 8 ] z 8 ] Im[z 12 z 6 z z 2 8 ] Im[z 10 z 12 z 6 z 2 8 ] Im[z 10 z 8 z z 2 z 4 ]
	115	0	3	Im[z 8 z 2 z 6 ]	166	0	5	Im[z 12 z 8 z z 4 z 6 ]
	116	0	3	Im[z 10 z 4 z 6 ]	167	0	5	Im[z 12 z 4 z z 6 z 8 ]
	117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0	3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5	Im[z 10 z 2 z 8 ] Im[z 12 z 4 z 8 ] Im[z 6 z 3 2 ] Im[z 12 z 3 4 ] Im[z 2 6 z 10 z 2 ] Im[z 2 8 z 12 z 4 ] Im[z 8 z 2 2 z 4 ] Im[z 10 z 2 z 2 4 ] Im[z 2 4 z 2 z 6 ] Im[z 10 z 2 2 z 6 ] Im[z 2 10 z 12 z 8 ] Im[z 12 z 2 2 z 8 ] Im[z 2 6 z 4 z 8 ] Im[z 2 8 z 10 z 6 ] Im[z 4 z 8 z 10 z 2 ] Im[z 10 z 4 z 12 z 2 ] Im[z 6 z 8 z 12 z 2 ] Im[z 6 z 8 z 10 z 4 ] Im[z 10 z 6 z 12 z 4 ] Im[z 10 z 8 z 12 z 6 ] Im[z 12 z 2 z 4 z 6 ] Im[z 4 z 6 z 2 z 8 ] Im[z 8 z 4 2 ] Im[z 2 6 z 3 4 ] Im[z 2 12 z 3 8 ] Im[z 3 4 z 10 z 2 ] Im[z 2 8 z 12 z 2 2 ] Im[z 2 12 z 2 10 z 4 ] Im[z 10 z 3 2 z 4 ] Im[z 2 10 z 12 z 2 4 ] Im[z 12 z 2 2 z 2 4 ] Im[z 12 z 3 2 z 6 ] Im[z 10 z 8 z 3 6 ] Im[z 2 6 z 2 2 z 8 ] Im[z 2 10 z 2 6 z 8 ] Im[z 2 10 z 4 z 2 8 ] Im[z 2 8 z 4 z 2 6 ] Im[z 2 4 z 6 z 12 z 2 ] Im[z 6 z 8 z 10 z 2 2 ] Im[z 10 z 6 z 12 z 2 2 ] Im[z 2 8 z 10 z 2 z 4 ] Im[z 12 z 6 z 10 z 2 4 ] Im[z 2 10 z 12 z 2 z 6 ] Im[z 10 z 4 z 2 z 2 6 ]	168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0	6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 9 9 9 10 11	Im[z 10 z 5 2 ] Im[z 2 8 z 10 z Im[z 3 8 z 2 10 z ] ] Im[z 12 z 4 2 z ] Im[z 3 z 6 ] 10 z 2 Im[z 3 ] 6 z 10 z Im[z 3 ] 8 z 12 z Im[z 2 ] 10 z 2 z Im[z 2 10 z 3 4 z ] Im[z 3 ] 6 z 2 z Im[z 10 z 6 z ] Im[z 6 z 2 8 z 10 z 2 ] Im[z 3 8 z 10 z z 2 ] Im[z 2 z 2 ] 10 z 6 z Im[z 10 z 8 z z 3 2 ] Im[z 10 z 2 z 2 ] 8 z Im[z 12 z 2 z 4 ] 6 z Im[z 2 z 4 ] 10 z 8 z Im[z 2 10 z z 2 2 z 4 ] Im[z 2 12 z z 2 4 z 6 ] Im[z 2 12 z 2 z z 2 8 ] Im[z 12 z 6 2 ] Im[z 3 8 z 4 6 ] Im[z 2 10 z 5 4 ] Im[z 2 10 z z 4 2 ] Im[z 4 6 z 2 10 z ] Im[z 3 z 6 ] 12 z 3 Im[z 3 10 z z 3 6 ] Im[z 3 12 z 2 z 2 8 ] Im[z 3 ] 10 z 6 z Im[z 2 10 z 12 z 4 8 ] Im[z 3 8 z 2 ] 10 z Im[z 2 10 z 8 z z 2 2 ] Im[z 3 z 2 z 4 ] 10 z 2 Im[z 3 10 z 5 6 ] Im[z 4 8 z 3 10 z ] Im[z 4 z 4 ] 10 z 3 Im[z 3 10 z 2 z 3 2 ] Im[z 3 z 2 ] 10 z 8 z Im[z 4 10 z 5 8 ] Im[z 4 10 z 3 z 2 2 ] Im[z 4 z 8 ] 12 z 4 Im[z 5 z 2 ] 10 z 4 Im[z 5 12 z 6 ]

Table 22 .

 22 Isotropic products of hemitropic invariants of S 12 (R 2 ) Im[z 10 z 2 z 8 ]

	#	order degree Formula
	110 111 112 113	0 0 0 0	6 6 6 6	Im[z 4 z 2 2 ] Im[z 12 z 2 6 ] Im[z 12 z 2 6 ] Im[z 8 z 2 4 ] Im[z 12 z 10 z 2 ] Im[z 8 z 2 4 ] Im[z 12 z 2 6 ]

  [T] = (T 111 , T 221 , T 122 , T 212 , T 222 , T 112 , T 211 , T 121 ) t , and the same for [ρ ′ (u)T] (as well as for the eigenvectors [W n k

  222 + 3T 112 -T 211 -T 121 ), 222 + T 112 + T 211 + T 121 ).Example A.6. Let V = Piez ⊂ T 3 (R 2 ) be the vector space of 2D piezoelectricity tensors P, where P ijk = P ikj . An orthonormal basis is given by The induced representation of the Lie algebra so(2) on Piez still writes as (A.3). We now set [P] = (P 111 , P 122 , √ 2P 212 , P 222 , P 211 , √ 2P 112 ) t , and the same for [ρ ′ (u)P] (and the [W n k ]), so that

		1 2	(T 111 + T 212 ) +	i 2	(T 222 + T 121 ),
	z 1b = T, W 1b =	1 √ 2	3	(T 111 + 2T 122 -T 212 ) -	i √ 2	3	(T 222 + 2T 211 -T 121 ),
	z 1c = T, W 1c = (T z 3 = T, W 3 = 1 2 √ 6 (T 111 + 3T 221 -T 122 -T 212 ) + i 2 √ 6 1 2 √ 2 (T 111 -T 221 -T 122 -T 212 ) + i 2 √ 2 (-T e 111 , e 122 , 1 √ 2 (e 212 + e 221 ), 1 √ 2 (e 112 + e 121 ),	e 211 ,	e 222 .

  and their three conjugates. This corresponds to an harmonic decomposition 2H1 (R 2 ) ⊕ H 3 (R 2 ) of Piez, where 222 + P 211 + 2P 112 ). t k z n k )(1t k z -n k ) z -1 dz = k 1 ,...,kr≥0 b k 1 ,...,kr t k 1 1 • • • t kr r ,and b k 1 ,...,kr = 1 2iπ0≤α i ≤k i |z|=1 z (2α 1 -k 1 )n 1 +•••+(2αr -kr)nr z -1 dzis the number of solutions (α 1 , . . . , α r ) of the linear Diophantine equation(C.2) 2α 1 n 1 + • • • + 2α r n r = k 1 n 1 + • • • + k r n r , α i ≥ 0. 1 ,...,kr≥0 β k 1 ,...,kr t k 1 1 • • • t kr r ,

	1 √ 6 ( √ 2, i, -i, -i √ 2) z 1a = T, W 1a = [W 3 ] = 1 2 √ 2 (1, -1, -√ 1 √ 3 (P 111 + P 212 ) + 2, 0, 1, -i √ √ 2, 0, -i) t , [W 1b ] = 1 2 √ i 3 (P 222 + P 112 ), z 1b = T, W 1b = 1 2 √ 6 (P 111 + 3P 122 -2P 212 ) + i 2 √ 6 (P 222 + 3P 211 -2P 112 ), 6 (1, 3, -√ 2, -i, -3i, i √ z 3 = T, W 2 = 1 2 √ 2 (P 111 -P 122 -2P 212 ) + i 2 √ 2 1 2πi |z|=1 r k=1 1 (1 Remark C.3. Note that (-P where 1 (1 -t 2 1 ) • • • (1 -t 2 r ) = ℓ 1 ,...,ℓr≥0 t 2ℓ 1 1 • • • t 2ℓr r =	2),

t k

  1 (1t k z n k )(1t k z -n k ) z -1 dz, Appendix D. ProofsIt is useful in invariant theory to introduce the following bi-differential operators△ αβ := ∂ 2 ∂x α ∂x β + ∂ 2 ∂y α ∂y β , Ω αβ := ∂ 2 ∂x α ∂y β -∂ 2 ∂y α ∂x β ,which are known respectively as the polarized Laplacian and the Cayley operator. The polarized Laplacian △ αβ is invariant under O(2) and the Cayley operator Ω αβ is invariant under SL(2, R). Anyway, they are both SO(2)-invariant. Since we shall use the complex variables (z, z) rather than the real variables (x, y), we introduce also the complex differential operators αβ + iΩ αβ ) ,or △ αβ = 2 D αβ + D βα , Ω αβ = -2i D αβ -D βα .

	∂ ∂z	=	1 2	∂ ∂x	-i	∂ ∂y		,	∂ ∂z	=	1 2	∂ ∂x	+ i	∂ ∂y	,
	and the complex bi-differential operator
						D αβ :=	∂ 2 ∂z α ∂z β	,
	and we get
					D αβ =	1 4	(△

Appendix A. An explicit harmonic decomposition

In this appendix, we propose a general method to obtain explicitly an harmonic decomposition of any linear representation V of the orthogonal groups SO [START_REF] Ashtekar | Geometrical formulation of quantum mechanics[END_REF] or O(2), the main application remaining the special case of a tensorial representation. The method exposed here is rather simple, since it requires only the diagonalization of a matrix. It is however limited to dimension 2 and cannot be generalized to dimension 3 or higher. This is not the only procedure to obtain

Appendix B. Harmonic decomposition of a totally symmetric tensor

The harmonic decomposition of an homogeneous polynomial p of degree n is unique and writes

where r := ⌊n/2⌋ (with ⌊x⌋ the floor function), q(x x x) := x 2 +y 2 and h k ∈ H n-2k (R 2 ) are harmonic polynomials of respective degree n -2k. These harmonic functions can be calculated iteratively using the relation

where h is an homogeneous harmonic polynomial of degree n and r ≥ 0. More precisely, we have

Then, we compute inductively h k , for k = r -1, r -2, . . . , using the relation

which leads to h 0 after r iterations. The harmonic function h 0 (of degree n) is called the leading harmonic part of p and will be denoted by p 0 when necessary. Using now the correspondence between totally symmetric tensors and homogeneous polynomials, we set h k = φ(H k ) and we get

We deduce then the harmonic decomposition of a totally symmetric tensor S ∈ S n (R 2 )

Moreover, using the identity φ(tr r S) = (n-2r)! n! ∆ r φ(S), valid for every symmetric tensor S or order n, we can explicit the factor H k , where we have set

and

The leading harmonic part S ′ := H 0 of S, such that φ(S ′ ) = φ(S) 0 , is an harmonic tensor in

Appendix C. The Hilbert series of an O(2)-representation Consider a linear representation (V, ρ) of a compact group G which splits into a direct sum of stable subspaces

where Inv k 1 ,...,kp (V, G) is the finite dimensional subspace of multi-homogeneous invariant polynomials p which have degree where a k 1 ,...,kp := dim Inv k 1 ,...,kp (V, G). The remarkable fact is that the Hilbert series is a rational function that can be calculated a priori, using the Molien-Weyl formula [START_REF] Sternberg | Group theory and physics[END_REF][START_REF] Sturmfels | Algorithms in Invariant Theory[END_REF].

Theorem C.1 (Molien-Weyl formula). The multi-graded Hilbert series (C.1) of Inv(V, G) writes as

where dµ is the Haar measure on G (see definition 5.7) and ρ k is the restriction of the representation to V k .

In the special case where G = O(2) and

which is the important case for us, we have the following more explicit result.

Theorem C.2. Let (V, ρ) be a linear representation of O(2) which decomposes as

,

Then, the Hilbert series of Inv(V, O(2)) writes as

, where

Setting k i := α i + β i , this latest sum can now be recast as

and we get thus

..,kr is the number of solutions of the linear Diophantine equation

Example C.6. Let Ela be the vector space of bidimensional elasticity tensors. Its harmonic decomposition (see example 3.5) writes

Let us associate to the components (λ, µ, h, H), the formal variables (t 0a , t 0b , t 2 , t 4 ). Then, using theorem C.2, remarks C.4 and C.5, we get

, and

.

The Taylor expansion of H (Ela,SO(2)) writes

We deduce from this expansion that the space of homogeneous invariants of multi-degree (1, 0, 0, 0), corresponding to t 0a , is of dimension 1 (and spanned by λ) and that the space of homogeneous invariants of multi-degree (0, 0, 2, 0) corresponding to t 2 2 is of dimension 1 (and spanned by tr a 2 ).

Besides, we have

for every symmetric tensor S or order n. Applying this formula to φ(S) = [H 1 , H 2 ], of degree n = N 1 + N 2 -2, with r = N 1 -1 and where

we get

and thus

which is the second identity of theorem 7.4. It remains to show that

To do so, we use the fact that Im z = Re(-iz) and thus that h k = Re(iz k z n k ). We have therefore

by (D.1) and (D.4). The same way, we deduce that

[106] A. Email address: rodrigue.desmorat@ens-paris-saclay.fr (Boris Kolev) Université Paris-Saclay, ENS Paris-Saclay, CentraleSupélec, CNRS, LMPS -Laboratoire de Mécanique Paris-Saclay, 91190, Gif-sur-Yvette, France

Email address: boris.kolev@math.cnrs.fr