Evelyne Hubert
email: evelyne.hubert@inria.fr

Tobias Metzlaff
email: tobias.metzlaff@rptu.de

Cordian Riener
email: cordian.riener@uit.no

Orbit spaces of Weyl groups acting on compact tori: a unified and explicit polynomial description

Keywords: Multiplicative Invariants, Weyl Groups, Root Systems, Orbit Spaces, Semi-Algebraic Sets, Polynomial Matrix Inequalities, Chebyshev Polynomials MSC: 13A50 14P10 17B22 33C52

The Weyl group of a crystallographic root system has a nonlinear action on the compact torus. The orbit space of this action is a compact basic semi-algebraic set. We present a polynomial description of this set for the Weyl groups associated to root systems of type A, B, C, D and G. Our description is given through a polynomial matrix inequality. The novelty lies in an approach via Hermite quadratic forms and a closed form formula for the matrix entries.

The orbit space of the nonlinear Weyl group action is the orthogonality region of generalized Chebyshev polynomials. In this polynomial basis, we show that the matrices obtained for the five types follow the same, surprisingly simple pattern. This is applied to the optimization of trigonometric polynomials with crystallographic symmetries.

Introduction

Subgroups of GL n (Z) define monomial (a.k.a. multiplicative) actions on the algebraic torus (C * ) n that leave the compact torus T n := {(x 1 , . . . , x n ) ∈ C n | ∀ 1 ≤ i ≤ n : |x i | = 1} invariant [START_REF] Lorenz | Multiplicative Invariant Theory[END_REF]. Of special significance among these groups are the integral representations of the Weyl groups. For the Weyl group W associated to a root system of type A n-1 , B n , C n , D n , or G 2 , we provide an explicit description of the orbit space T n /W. It is a compact semi-algebraic set that is given as the locus of positive semi-definiteness of a Hankel matrix with polynomial entries. We give closed form formulae for the entries of these matrices, initially in a monomial basis (Theorems 5.3, 6.3, 7.4, 8.4 and 9.3). In Figure 1 below we depict the resulting semialgebraic sets for the two and three dimensional cases. Remarkably, for the cases cited above, these matrices have a uniform formula when their entries are expressed in terms of the generalized Chebyshev polynomials associated to the root system (Theorem 10.1).

A2 C2 B2 G2 A3 C3 B3
Figure 1: The T-orbit space for the irreducible Weyl groups of rank 2 and 3.

We can draw an analogy of our problem with the determination of the orbit space R n /G of the linear action of a compact group G on R n . Explicit formulae of the semi-algebraic set representing R n /G are also given for the (signed) permutation representations of the symmetric group S n , S n ⋉ {±1} n and S n ⋉ {±1} n-1 [START_REF] Procesi | Positive symmetric functions[END_REF][START_REF] Talamini | Generating formulas for finite reflection groups of the infinite series Sn, An, Bn and Dn[END_REF]. These formulae are special cases of the encompassing construction of real orbit spaces given in [START_REF] Procesi | Inequalities defining orbit spaces[END_REF]. Though the generalization outside of linear actions presented in [PS85, Section 4] could be applied to our cases, our approach is different and its main ingredients from real algebraic geometry are outlined below.

In both approaches the orbit space is given as the image of T n by a fundamental set of invariants. We show that a point z is in this image if and only if all the solutions of a certain polynomial system belong to T n . The polynomial system is given in terms of the elementary symmetric functions and thus, by Vieta's theorem, the problem boils down to characterizing when all the roots of a univariate polynomial belong to the interval [-1, 1]. Our Hankel matrix stems from the application of Sturm's theorem [START_REF] Cox | Using algebraic geometry[END_REF] (a.k.a. Sylvester inertia principle [START_REF] Procesi | Positive symmetric functions[END_REF]).

The Weyl groups W associated to the root systems of type A n-1 , B n , C n , or D n are semi-direct products S n ⋉ {±1} k , where S n is the symmetric group and k is respectively 0, n, n or n -1; For G 2 , (n, k) = (3, 1). The first idea of our constructions is to write the elementary symmetric functions, in some invariants of the monomial action of {±1} k , in terms of a fundamental set of invariants for the whole Weyl group (and not the contrary). Several sets of fundamental invariants are known explicitely [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines[END_REF]. The treatment for C n is the most straightforward and the other cases treated derived from there. The Weyl groups associated with the remaining exceptional (crystallographic) root systems, F 4 , E 6 , E 7 , and E 8 , do not entertain a simple decomposition in terms of symmetric group and hence we do not forsee how to apply our approach there.

Our original motivation was to propose a symmetry reduction scheme for the optimization of trigonometric polynomials, as they appear in bounding the chromatic number of some infinite Cayley graphs [START_REF] Hubert | Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs[END_REF].

Yet the regions we describe as full dimensional semi-algebraic sets already appeared as of interest in analysis.

For instance they are the only Gaussian cubature domains known in all dimensions and all degrees [LSX08, LX10, MP11, LSX12, Xu15, HHM16]. The nodes of these cubatures are known explicitely and provide suitable sampling points [START_REF] Munthe-Kaas | Through the kaleidoscope: symmetries, groups and Chebyshev-approximations from a computational point of view[END_REF]. This is all connected to the fact that these are the regions of orthogonality of the generalized Chebyshev polynomials [START_REF] Hoffman | Generalized Chebyshev polynomials associated with affine Weyl groups[END_REF]. These polynomials have entered the area of multivariate special functions with the work of Koornwinder [START_REF] Koornwinder | Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators[END_REF], MacDonald [START_REF] Macdonald | Orthogonal polynomials associated with root systems[END_REF] and beyond [START_REF] Dunn | Multi-dimensional generalizations of the Chebyshev polynomials, I, II[END_REF][START_REF] Eier | A class of orthogonal polynomials in k variables[END_REF][START_REF] Beerends | Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator[END_REF]. In this paper, we only draw on their definition through the ring of invariants of the Weyl groups.

Yet their interpretation via discrete Fourier analyis based on full dimensional lattices with symmetry should entail further analytical interest.

This article is organized as follows. In Section 2 we introduce, through an example, the general context in which we were drawn to investigate the problem solved here: optimization of trigonometric polynomials with symmetry. Then Section 3 contains mostly standard material from the theory of root systems and lattices from [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines[END_REF] and helps us settling the notations. The following Section 4 is our toolbox for characterizing when all solutions of the polynomial systems we shall encounter are contained in the compact torus. These tools are based on the material in [START_REF] Cox | Using algebraic geometry[END_REF]. In Sections 5 to 9 we treat in turn the cases A n-1 , C n , B n , D n , and G 2 in a monomial basis. Finally, in Section 10, we give the unified formula for all cases in their respective Chebyshev basis.

The Maple package GeneralizedChebyshev1 supports all the constructions and formulae we present in this paper. In particular all the computations and the figures in this article were produced with this package, and the plotting tools of Maple.

Motivation

Let Ω be a full-dimensional lattice in a finite-dimensional real vector space V with inner product ⟨•, •⟩.

Denote by Λ the dual lattice of Ω and let f :

V → R be an L 2 -integrable Λ-periodic function. Then f has a Fourier expansion f : V → R, u → µ∈Ω f µ exp(-2πi ⟨µ, u⟩), (2.1) with coefficients f µ = f -µ ∈ C, see [DM85, CS99
]. Since f is continuous and the periodicity domain V /Λ is compact, the global minimum

f * := min u∈V /Λ f (u) (2.2)
exists. If all but finitely many coefficients f µ are zero, then we call f a trigonometric polynomial.

Trigonometric polynomials are good L 2 -approximations for Λ-periodic functions [START_REF] Munthe-Kaas | Through the kaleidoscope: symmetries, groups and Chebyshev-approximations from a computational point of view[END_REF]. The trigonometric optimization problem is to find the value f * and arises, for example, in filter design [START_REF] Dumitrescu | Positive Trigonometric Polynomials and Signal Processing Applications[END_REF], computation of spectral bounds [START_REF] Bachoc | Spectral bounds for the independence ratio and the chromatic number of an operator[END_REF], and optimal power flow [START_REF] Bai | Semidefinite programming for optimal power flow problems[END_REF][START_REF] Josz | Lasserre hierarchy for large scale polynomial optimization in real and complex variables[END_REF].

In joint work with Philippe Moustrou [START_REF] Hubert | Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs[END_REF], the authors address the trigonometric optimization problem for trigonometric polynomials with Weyl group symmetry. The idea is to rewrite f as a classical polynomial on the T-orbit space of the multiplicative Weyl group action on V /Λ. Having an explicit description of this orbit space as a basic semi-algebraic set allows us to use the sums of square/moment hierarchy for global optimization [START_REF] Lasserre | Global Optimization with Polynomials and the Problem of Moments[END_REF][START_REF] Laurent | Sums of Squares, Moment Matrices and Optimization Over Polynomials[END_REF][START_REF] Blekherman | Semidefinite optimization and convex algebraic geometry[END_REF].

Example 2.1. Let Ω = Z 2 be the square lattice in V = R 2 and consider the trigonometric polynomial

f (u) := exp(-2πi ⟨± 1 0 , u⟩) + exp(-2πi ⟨± 0 -1 , u⟩) + 1 2 exp(-2πi ⟨± 1 1 , u⟩) + exp(-2πi ⟨± 1 -1 , u⟩) -exp(-2πi ⟨± 2 2 , u⟩) -exp(-2πi ⟨± 2 -2 , u⟩) - 3 4 exp(-2πi ⟨± 2 1 , u⟩) + exp(-2πi ⟨± 2 -1 , u⟩) + exp(-2πi ⟨± 1 2 , u⟩) + exp(-2πi ⟨± 1 -2 , u⟩) .
The group W ∼ = S 2 ⋉ {±1} 2 acts on R 2 by permutation and sign change of coordinates. Note that f is invariant under the substitution u → s(u) whenever s ∈ W.

By Bourbaki's Theorem 3.4, the set of trigonometric polynomials, which are left invariant under this substitution, is a polynomial algebra. This means that we can fix a set of fundamental invariants {θ 1 , θ 2 } and a map ϑ : u → (θ 1 (u), θ 2 (u)), so that f (u) can be rewritten uniquely as a polynomial g(ϑ(u)). In particular, the minimum is

f * = min u∈R 2 f (u) = min u∈R 2 g(ϑ(u)) = min z∈ϑ(R 2 ) g(z) = min z 16 z 2 1 -12 z 1 z 2 -8 z 2 2 + 10 z 1 -6 z 2 -2, s.t. z = (z 1 , z 2 ) ∈ R 2 with -4 z 2 1 + 2 z 2 + 2 -8 z 3 1 + 6 z 1 z 2 + 2 z 1 -8 z 3 1 + 6 z 1 z 2 + 2 z 1 -16 z 4 1 + 16 z 2 1 z 2 + 4 z 2 1 -2 z 2 2 -2 z 2 ⪰ 0,
where ⪰ 0 means that the matrix is positive semi-definite. The rewriting process from f (u) to g(z) = 8 z 2 1 -. . . is straightforward once a set of fundamental invariants is fixed, see Example 3.7. The matrix is obtained with our formula in Theorem 6.3 and describes the new feasible region in Figure 2.

The T-orbit space of a Weyl group

A key theorem from multiplicative invariant theory states that, given a finite group of integer matrices with a multiplicative action, the invariant ring is a polynomial algebra if and only if the group is a Weyl group. Classical references for this statement are the book of Bourbaki [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines[END_REF] and the papers by Steinberg [START_REF] Steinberg | On a theorem of Pittie[END_REF] and Farkas [START_REF] Farkas | Multiplicative invariants[END_REF]. In this section, we fix the terminology and notation: We revisit the theory of Weyl groups from [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines[END_REF] and of their multiplicative actions on compact tori from [START_REF] Lorenz | Multiplicative Invariant Theory[END_REF]. This admits a definition of generalized Chebyshev polynomials, which will later turn out to be the suitable basis for our explicit description of the associated orbit space. Finally, we show that the orbit space can always be assumed to be a real compact set.

T-orbit spaces

We recall the theory of multiplicative invariants from [START_REF] Lorenz | Multiplicative Invariant Theory[END_REF]. Let V be a finite-dimensional R-vector space with inner product ⟨•, •⟩ :

V × V → R, W be a finite subgroup of GL(V ) and Ω ⊆ V be a full-dimensional W-invariant lattice, that is, Ω ⊗ Z R = V and W Ω = Ω.
We fix a Z-basis {ω 1 , . . . , ω n } for Ω and denote by φ : Z n → Ω the Z-module isomorphism that takes the standard basis vector e i to ω i . Then W can be represented as W ⊆ GL n (Z), where the group elements s ∈ W are replaced by matrices B: The action on a column vector α ∈ Z n is given by matrix vector multiplication

B α := φ -1 (s(φ(α))) = α 1 s(ω 1 ) + . . . + α n s(ω n ).
Denote by Λ := {λ ∈ V | ∀ µ ∈ Ω : ⟨µ, λ⟩ ∈ Z} the dual lattice of Ω. The periodicity domain of a trigonometric polynomial as in Equation (2.1) is the compact additive group V /Λ. Furthermore, the compact n-torus

T n := {x = (x 1 , . . . , x n ) ∈ C n | ∀ 1 ≤ i ≤ n : |x i | = 1} is a multiplicative group. A group isomorphism between V /Λ and T n is given by u + V → (exp(-2πi ⟨ω i , u⟩)) 1≤i≤n .
The complexification of T n is the algebraic n-torus (C * ) n := (C \ {0}) n . For x = (x 1 , . . . , x n ) ∈ (C * ) n and a column vector α = [α 1 , . . . , α n ] t ∈ Z n , we define x α := x α1 1 . . . x αn n ∈ C * . Then

⋆ : W × (C * ) n → (C * ) n , (B, x) → B ⋆ x := x B -1 := (x B -1 •1 , . . . , x B -1 •n ) (3.1)
is a nonlinear action, where B -1 •i ∈ Z n denotes the i-th column vector of the inverse matrix B -1 ∈ W. The orbits are denoted by W ⋆ x and the action ⋆ leaves the compact n-torus invariant, that is W ⋆ T n = T n . We refer to ⋆ as the multiplicative action of W on the (compact) torus.

The coordinate ring of (C * ) n is the multivariate Laurent polynomial ring

C[x ± ] := C[x 1 , x -1 1 , . . . , x n , x -1 n ]. The monomials in C[x ± ] are the x α = x α1
1 . . . x αn n with α ∈ Z n and form the multiplicative unit group of C[x ± ]. The nonlinear action ⋆ induces a linear action on C[x ± ], given by its values on the monomial basis

• : W × C[x ± ] → C[x ± ], (B, x α ) → B • x α := x Bα . (3.2) For f = α f α x α ∈ C[x ± ], B ∈ W and x ∈ (C * ) n , we have (B • f )(x) = α f α x Bα = f (B -1 ⋆ x). The orbits are denoted by W • f . If W • f = {f }, then f is called W-invariant. The set of all W-invariant Laurent polynomials is a subring denoted by C[x ± ] W . The orbit polynomial associated to α ∈ Z n is Θ α := 1 |W| B∈W x Bα ∈ C[x ± ] W . (3.3) If α ∈ Wα, then Θ α = Θ α .
T n /W → C m , W ⋆ x → (θ 1 (x), . . . , θ m (x))
is well-defined and injective.

Proof. For x, y ∈ T n with W ⋆ x = W ⋆ y and 1 ≤ i ≤ m, we have θ i (x) = θ i (y) by definition. Therefore, the map is well-defined.

For injectiveness, we follow the proof of [CLO15, Ch. 7, §4, Thm. 10] for linear actions: Assume that

x, y ∈ T n with W ⋆ x ̸ = W ⋆ y. Then W ⋆ x ∩ W ⋆ y = ∅ and X := W ⋆ x ∪ W ⋆ y \ {y} ⊆ T n is a finite set.
We define a polynomial f :=

x ′ ∈X n i=1 (x i -x ′ i ) ∈ C[x 1 , . . . , x n ],
so that f (X) = {0} and f (y) ̸ = 0. By letting W act linearly on f as a Laurent polynomial, we may define

f := 1 |W| B∈W B • f ∈ C[x ±1 1 , . . . , x ±1 n ] W .
Since {θ 1 , . . . , θ m } is a set of fundamental invariants, there exists a polynomial g ∈ C[z 1 , . . . , z m ], such that

g(θ 1 (x), . . . , θ m (x)) = f (x) = 0 ̸ = f (y)/|W ⋆ y| = f (y) = g(θ 1 (y), . . . , θ m (y)).
In particular, we have (θ 1 (x), . . . , θ m (x)) ̸ = (θ 1 (y), . . . , θ m (y)). □

For a fixed set {θ 1 , . . . , θ m } ⊆ C[x ± ] W of fundamental invariants, we define the associated map

ϑ : (C * ) n → C m , x → (θ 1 (x), . . . , θ m (x)).
(3.5) Definition 3.2. We call T := ϑ(T n ) the T-orbit space of the multiplicative action of W on T n w.r.t. ϑ.

Root systems and Weyl groups

The groups and lattices that we are interested in arise from root systems. We take the definitions from [Bou68, Ch. VI]. Again, V is a finite-dimensional R-vector space with inner product ⟨•, •⟩ :

V × V → R. A subset R ⊆ V is called a root system in V , if the following conditions hold.
R1 The set R is finite, spans V as an R-vector space and does not contain 0.

R2 If ρ, ρ ∈ R, then ⟨ρ, ρ ∨ ⟩ ∈ Z, where ρ ∨ := 2 ρ ⟨ρ,ρ⟩ . R3 If ρ, ρ ∈ R, then s ρ (ρ) ∈ R, where s ρ is the reflection s ρ (u) := u -⟨u, ρ ∨ ⟩ρ for u ∈ V .
Furthermore, R is said to be reduced, if the following condition holds.

R4 For ρ ∈ R and c ∈ R, we have cρ ∈ R if and only if c = ±1.

We assume that the "reduced" property R4 is always fulfilled when we speak of a "root system". The distinction between root systems and roots of polynomials is assumed to be clear from the context.

The Weyl group W of R is the group generated by the reflections s ρ for ρ ∈ R. This is a finite subgroup of the orthogonal group on V with respect to the inner product ⟨•, •⟩. The orbits are denoted by W u for u ∈ V . The rank of R, respectively W, is the dimension of V .

A weight of R is an element µ ∈ V , such that, for all ρ ∈ R, we have ⟨µ, ρ ∨ ⟩ ∈ Z. The weight lattice Ω is the set of all weights and a full-dimensional W-lattice.

Let dim(V ) = n and B = {ρ 1 , . . . , ρ n } ⊆ R be a subset with the following properties.

B1 The set B is an R-vector space basis for V .

B2 For every ρ ∈ R, there exists α ∈ N n , such that ρ = α 1 ρ 1 + . . .

+ α n ρ n or ρ = -α 1 ρ 1 -. . . -α n ρ n .
Such a set B always exists and is called base [Bou68, Ch. VI, §1, Thm. 3]. We define the fundamental weights as the linearly independent ω 1 , . . . , ω n ∈ Ω, such that, for 1 ≤ i, j ≤ n, we have ⟨ω i , ρ ∨ j ⟩ = δ i,j .

Proposition 3.3. Denote by S n the symmetric group on n points. There exists a permutation σ ∈ S n of order at most 2, such that, for all 1 ≤ i ≤ n, we have -ω i ∈ W ω σ(i) .

Proof. The statement follows from the existence of the longest Weyl group element s 0 ∈ W: For the proof, it suffices to know that s 0 is a reflection that takes a fixed base {ρ 1 , . . . ,

ρ n } to {-ρ 1 , . . . , -ρ n }, see [Bou68,
Ch. VI, §1, Prop. 17, Coro. 3]. Hence, there is a permutation σ ∈ S n with s 0 (ρ i ) = -ρ σ(i) . Since s 2 0 = Id V and the inner product is W-invariant, σ is of order at most 2 and thus

-s 0 (ω i ) = n j=1 ⟨-s 0 (ω i ), ρ ∨ j ⟩ ω j = n j=1 ⟨ω i , -s 0 (ρ ∨ j )⟩ ω j = n j=1 ⟨ω i , ρ ∨ σ(j) ⟩ ω j = ω σ(i)
is also a fundamental weight.

□ Assume that V = V (1) ⊕ . . . ⊕ V (k)
is the direct sum of proper orthogonal subspaces and that, for each

1 ≤ i ≤ k, there is a root system R (i) in V (i) . Then R (1) ∪ . . . ∪ R (k) is a root system in V called the direct sum of the R (i) . If

Chebyshev polynomials

For the remainder of this article, W is the Weyl group of a root system. We fix a set of fundamental weights {ω 1 , . . . , ω n } as a Z-basis of the weight lattice Ω. Thus, we may write W ⊆ GL n (Z). We already know that C[x ± ] W is finitely generated, but we need to fix a set of generators in order to define the T-orbit space.

Theorem 3.4. [Bou68, Ch. VI, §3, Thm. 1] The orbit polynomials Θ e1 , . . . , Θ en are algebraically independent and {Θ e1 , . . . ,

Θ en } ⊆ C[x ± ] W is a set of fundamental invariants.
For 1 ≤ i ≤ n, we denote by θ i := Θ ei from now on the orbit polynomial associated to e i ∈ Z n . Let ϑ be the associated map from Equation (3.5) with n = m. By Proposition 3.1, the T-orbit space T = ϑ(T n ) of the multiplicative action of W on T n w.r.t. ϑ is in bijection to T n /W. Since the θ i are algebraically independent, the image T does not lie in any proper subvariety and is full-dimensional in an n-dimensional R-vetor space, that we specify later.

To describe T , it suffices to consider irreducible root systems. Indeed, the Weyl group W of a root system R is the outer product of the Weyl groups corresponding to the irreducible components, see the discussion before [Bou68, Ch. VI, §1,

Prop. 5]. Hence, if R = R (1) ∪ . . . ∪ R (k) is a direct sum of irreducible root systems R (i) , then the T-orbit space is a Cartesian product T = T (1) × . . . × T (k) of T-orbit spaces T (i) . By Theorem 3.4, C[x ± ] W = C[θ 1 , . . . , θ n
] is a polynomial algebra. Hence, for every α ∈ Z n , there is a unique polynomial T α ∈ C[z 1 , . . . , z n ], so that the orbit polynomial Θ α can be written as

Θ α (x) = T α (θ 1 (x), . . . , θ n (x)). (3.6)
The polynomial T α can be computed recursively via the recurrence formula from Equation (3.4). Note that the monomials occurring in Θ α all have the same coefficient, namely 1/|Wα| ∈ Q. In particular, T α has rational coefficients. To describe the T-orbit space, we can therefore work over the multivariate polynomial ring

Q[z] := Q[z 1 , . . . , z n ].
Definition 3.5. The generalized Chebyshev polynomial of the first kind associated to α ∈ Z n is the unique

T α ∈ Q[z] from Equation (3.6).
For every α ∈ Z n , there exists a unique

α ∈ N n , such that α ∈ Wα [Bou68, Ch. VI, §1, Thm. 2]. In particular, {T α | α ∈ N n } is a Q-vector space basis for Q[z]
. By the definition, we have T 0 = 1, T e1 = z 1 , . . . , T en = z n . For higher degrees, T α depends on the type of the root system.

Example 3.6. For n = 1, there is only one integral Weyl group representation, namely W = {±1}. Thus,

T ℓ (x + x -1 )/2 = (x ℓ + x -ℓ )/2 defines the univariate Chebyshev polynomial associated to ℓ ∈ N.
The generalized Chebyshev polynomials of the first kind for type A 2 are

T 02 = 3 z 2 2 -2 z 1 , T 11 = 3/2 z 1 z 2 -1/2, T 20 = 3 z 2 1 -2 z 2 , T 03 = 9 z 3 2 -9 z 1 z 2 + 1, T 12 = 9/2 z 1 z 2 2 -3 z 2 1 -1/2 z 2 , T 21 = -3 z 2 2 -1/2 z 1 + 9/2 z 2 1 z 2 , T 30 = 9 z 3 1 -9 z 1 z 2 + 1, . . .
The generalized Chebyshev polynomials of the first kind for type C 2 are

T 02 = -8 z 2 1 + 4 z 2 2 + 4 z 2 + 1, T 11 = 2 z 1 z 2 -z 1 , T 20 = 4 z 2 1 -2 z 2 -1, T 03 = -48 z 2 1 z 2 + 16 z 3 2 + 24 z 2 2 + 9 z 2 , T 12 = -16 z 3 1 + 8 z 1 z 2 2 + 6 z 1 z 2 + 3 z 1 , T 21 = 8 z 2 1 z 2 -4 z 2 2 -3 z 2 , T 30 = 8 z 2 1 z 2 -4 z 2 2 -3 z 2 , . . .
Example 3.7. In Example 2.1, we claim that the rewriting process from a trigonometric polynomial with Weyl group symmetry to a classical one is straightforward: From the classification in [Bou68, Planches I-IX], we know that the group W ∼ = S 2 ⋉ {±1} 2 is the Weyl group of a root system in V = R 2 of type C 2 with weight lattice Ω = Z 2 and fundamental weights

ω 1 = [1, 0] t and ω 2 = [1, 1] t . Now, after substitution u → x := (x 1 , x 2 ) := (exp(-2πi ⟨ω 1 , u⟩), exp(-2πi ⟨ω 2 , u⟩
)), we redefine f to be

f := x ±e1 + x ±(e1-e2) + 1 2 x ±e2 + x ±(2 e1-e2) -x ±2 e2 -x ±(4 e1-2 e2) - 3 4 x ±(e1+e2) + x ±(3 e1-e2) -x ±(-e1+2 e2) -x ±(3 e1-2 e2) ∈ C[x ± ] W .
We collect the terms with exponent α ∈ N 2 . The remaining exponents are contained in the orbit Wα and thus occur as exponents in the associated orbit polynomial Θ α . Specifically,

f = x e1 + 1 2 x e2 -x 2 e2 - 3 4 x e1+e2 + . . . =|W e 1 |Θ e1 (x) + |W e 2 | 2 (Θ e2 (x) -Θ 2 e2 (x)) - 3|W (e 1 + e 2 )| 4 Θ e1+e2 = 4 Θ e1 (x) + 2 (Θ e2 (x) -Θ 2 e2 (x)) -6 Θ e1+e2 (x).
With Θ α (x) = T α (ϑ(x)) and z := ϑ(x), the minimum is

f * = min x∈T 2 f (x) = min z∈T 4 T e1 (z)+2 (T e2 (z) -T 2 e2 (z))-2 T e1+e2 (z) = min z∈T 16 z 2 1 -12 z 1 z 2 -8 z 2 2 +10 z 1 -6 z 2 -2.
The new feasible region T is the T-orbits space of the multiplicative action of S 2 ⋉ {±1} 2 on T 2 w.r.t. ϑ.

Real embedding

We want to describe T as a compact semi-algebraic set in R n . For x ∈ T n and 1 ≤ i ≤ n, we observe

θ i (x) = θ i (-I n ⋆ x) = (-I n • θ i )(x) = θ σ(i) (x), (3.7)
where σ ∈ S n is the permutation from Proposition 3.3. Hence, if -I n / ∈ W ⊆ GL n (Z), or equivalently, if σ is not trivial, then T is not a real set as the image of T n under x → ϑ(x) = (θ 1 (x), . . . θ n (x)). The irreducible root systems, for which this is the case, are both A n-1 and D 2n-1 whenever n ≥ 3 as well as E 6 .

We fix this circumstance in a straightforward manner: When j = σ(j), we set θ j := θ j ∈ C[x ± ]. When j < σ(j), we replace the j-th, respectively σ(j)-th, coordinate of ϑ by θ

j := (θ j + θ σ(j) )/2 ∈ C[x ± ], respectively θ σ(j) := (θ j -θ σ(j) )/(2i) ∈ C[x ± ]. For x ∈ T n , we have θ j (x) = ℜ(θ j (x)) ∈ R and θ σ(j) (x) = ℑ(θ j (x)) ∈ R.
Thus, the image of the map ϑ :

T n → R n , x → θ 1 (x), . . . , θ n (x) (3.8) is contained in the cube [-1, 1] n ⊆ R n .
We denote this image by T and call it the real T-orbit space of the multiplicative action of W on T n . On T , we define the real generalized Chebyshev polynomials.

Proposition 3.8. Let α, α ∈ N n with -α ∈ W α. There exist unique polynomials T α , T α ∈ Q[ z], such that T α (ϑ(x)) = T α ( ϑ(x)) + i T α ( ϑ(x)) and T α (ϑ(x)) = T α ( ϑ(x)) -i T α ( ϑ(x)). Proof. Assume that T α = β c β z β ∈ Q[z] for some c β ∈ Q and β ∈ N n . For x ∈ T n , we observe that T α (ϑ(x)) = β c β n j=1 (ℜ(θ j (x)) + i ℑ(θ j (x))) βj and T α (ϑ(x)) = β c β n j=1 ℜ(θ σ(j) (x)) + i ℑ(θ σ(j) (x)) βj are complex conjugates.
Furthermore, if j = σ(j), then ℜ(θ j (x)) = θ j (x) and ℑ(θ j (x)) = 0. Otherwise, our definition of θ j,R implies

ℜ(θ j (x)) = θ j (x), if j < σ(j) θ σ(j) (x), if j > σ(j) and ℑ(θ j (x)) = θ σ(j) (x), if j < σ(j) -θ j (x), if j > σ(j) .
Altogether, we obtain

Tα(ϑ(x))+T α (ϑ(x)) 2 = β c β 2 j=σ(j) θ j (x) βj j<σ(j) θ j (x) + i θ σ(j) (x) βj θ j (x) -i θ σ(j) (x) β σ(j) + j<σ(j) θ j (x) -i θ σ(j) (x) βj θ j (x) + i θ σ(j) (x) β σ(j)
.

The right hand side is a unique polynomial in ϑ(x) = ( θ 1 (x), . . . , θ n (x)), denoted by T α . Since the left hand side is real for every x ∈ T n , the coefficient of T α in front of i must be 0. Hence, we have

T α ∈ Q[z]. Similarly, by computing (T α -T α )/(2i), we obtain T α ∈ Q[z]. □ Example 3.9. If R is of type B n , C n or G 2 , then T =
T and so it is not necessary to distinguish.

The same holds for

D n , if n is even. If n is odd, then T k ei (z) = T k ei ( z) with zn-1+zn 2 = z n-1 , zn-1-zn 2i = z n .
For A n-1 , we have

T k e i (z)+T -k e i (z) 2 = T k e i (z)+T k e n-i (z) 2 = T k ei ( z) with zn-i+zi 2 = z i , zn-i-zi 2i = z n-i .
Specifically, for type A 2 with Weyl group W ∼ = S 3 , we have -e 1 ∈ W e 2 ⊆ Z 2 and the generalized Chebyshev polynomials of degree 2 are

T 02 = 3 z 2 2 -2 z 1 , T 11 = 3/2 z 1 z 2 -1/2, T 20 = 3 z 2 1 -2 z 2 ∈ Q[z].
After substitution

z 1 → z 1 + i z 2 , z 2 → z 1 -i z 2 , we obtain T 20 , T 02 = (3 z 2 1 -3 z 2 2 -2 z 1 ) ± i (6 z 1 z 2 + 2 z 2 ), T 11 = (3/2 z 2 1 + 3/2 z 2 2 -1/2),
and thus the new polynomials from Proposition 3.8 are

T 02 = 6 z 1 z 2 + 2 z 2 , T 11 = 3/2 z 2 1 + 3/2 z 2 2 -1/2, T 20 = 3 z 2 1 -3 z 2 2 -2 z 1 .
Those assume real values on T , which is the "deltoid" depicted in Figure 5 as a subset of R 2 .

Symmetric polynomial systems

In order to give a polynomial description for the T-orbit space T of a Weyl group W, our first step is to characterize when a symmetric polynomial system has a solution in the compact torus. The present section should be regarded as a toolbox for the proofs of the main results.

For 1 ≤ i ≤ n, we denote by σ i (y 1 , . . . , y n ) = |K|=i k∈K y k the i-th elementary symmetric function in n indeterminates y i , where K ranges over all subsets of {1, . . . , n} with cardinality i. We shall be confronted with the following two types of polynomial systems.

(I) σ i (y 1 , . . . , y n ) = (-1) i c i for 1 ≤ i ≤ n with c 1 , . . . , c n ∈ C (II) σ i y1+y -1 1 2 , . . . , yn+y -1 n 2 = (-1) i c i for 1 ≤ i ≤ n with c 1 , . . . , c n ∈ R
The goal is to determine, whether all complex solutions y = (y 1 , . . . , y n ) of system (I), respectively (II), are contained in the compact torus T n .

We pursue this goal, because the above situation arises in Sections 5 to 10. Later, the y i will assume the role of multivariate monomials in C[x ± ], such that y(x) ∈ T n if and only if z = ϑ(x) ∈ T .

Solutions in the compact torus

For now, x is a univariate indeterminate. Recall Vieta's formula

n k=1 (x-y k ) = x n + n i=1 (-1) i σ i (y 1 , . . . , y n ) x n-i . Lemma 4.1. 1. The map C * → C, x → x+x -1
2 is surjective and the preimage of the real interval [-1, 1] ⊆ R is T.

2. System (I) has a unique solution y ∈ C n up to permutation of its coordinates y i .

3. System (II) has a unique solution y ∈ (C * ) n up to permutation and inversion of its coordinates y i .

Proof. 1. For r ∈ C, consider the univariate polynomial p := x 2 -2 r x + 1. Then 0 is not a root of p and we have p(x) = 0 if and only if r = (x + x -1 )/2, that is, r is in the image of the map. If r ∈ [-1, 1], then the discriminant of p is r 2 -1 ≤ 0 and the two roots are x, x = x, x -1 = r ± i √ 1 -r 2 ∈ T. On the other hand, for x ∈ T, we have

(x + x -1 )/2 = (x + x)/2 = ℜ(x) ∈ [-1, 1].
2. By Vieta's formula, a solution y of system (I) is the vector of roots of the polynomial x n +c 1 x n-1 +. . .+c n . Since C is algebraically closed, the statement follows.

3. By 1., we can write the roots r 1 , . . . , r n of the polynomial x n + c 1 x n-1 + . . . + c n as r i = (y i + y -1 i )/2 for some y ∈ (C * ) n . Then y is a unique solution of (II) up to permutation and inversion.

□

From now on, we speak of "the" solution of system (I), respectively (II).

Proposition 4.2. For c 1 , . . . , c n ∈ R, the corresponding solution of system (II) is contained in T n if and only if all the roots of the univariate polynomial

x n + c 1 x n-1 + . . . + c n-1 x + c n are contained in [-1, 1].
Proof. If y ∈ (C * ) n is the solution of system (II), then the roots of x n +c 1 x n-1 +. . .+c n are r i := (y i +y -1 i )/2. According to Lemma 4.1, we have y ∈ T n if and only if r 1 , . . . , r n ∈ [-1, 1]. □

Similarly, we can characterize solutions of system (I). Recall that the univariate Chebyshev polynomial of the first kind associated to ℓ ∈ N is the unique

T ℓ with T ℓ ((x + x -1 )/2) = (x ℓ + x -ℓ )/2.
Proposition 4.3. For c 1 , . . . , c n-1 ∈ C with c i = (-1) n c n-i and c 0 := (-1) n c n := 1, the corresponding solution of system (I) is contained in T n if and only if all the roots of the univariate polynomial

T n (x) + d 1 T n-1 (x) + . . . + d n-1 T 1 (x) + d n 2 T 0 (x) with d ℓ = ℓ i=0 c i c ℓ-i ∈ R are contained in [-1, 1].
Proof. The solution of system (I) is contained in T n if and only if all the roots of the univariate polynomial p := x n + c 1 x n-1 + . . . + c n are contained in T.

The roots of p are nonzero, because p(0) = c n = (-1) n ̸ = 0. We define another univariate polynomial p := x n + c 1 x n-1 + . . . + c n . By the hypothesis, we have p(x) = (-x) n p(x -1 ). Hence, the set of all roots of pp consists of the roots of p and their inverses. In particular, all the roots of p are contained in T if and only if this holds for all the roots of pp.

We now show that pp is the polynomial from the statement: For 0 ≤ ℓ ≤ n, the coefficients of pp satisfy

Coeff(x ℓ , pp) = ℓ i=0 c n-i c n-ℓ+i = ℓ i=0 c i c ℓ-i = ℓ i=0 Coeff(x n-i , p) Coeff(x n-ℓ+i , p) = Coeff(x 2n-ℓ , pp).
The coefficients are real, because

ℓ i=0 c i c ℓ-i = ⌊(ℓ-1)/2⌋ i=0 (c i c ℓ-i + c ℓ-i c i ) ∈R + c ℓ/2 c ℓ/2 , ℓ even 0, ℓ odd ∈ R.
Thus, Coeff(x ℓ , pp) = Coeff(x 2n-ℓ , pp) = d ℓ ∈ R and we can write

pp = n ℓ=1 d n-ℓ (x n+ℓ + x n-ℓ ) + d n = 2x n n ℓ=1 d n-ℓ T ℓ x + x -1 2 + d n 2 T 0 x + x -1 2 =: 2x n g x + x -1 2 .
With Lemma 4.1, we see that x ∈ T is a root of pp if and only if (x + x -1 )/2 ∈ [-1, 1] is a root of g. □

The following example illustrates the proof and reappears in the example in Section 5.

Example 4.4. For c ∈ C, set p = x 3 -c x 2 + c x -1 and consider the palindromic polynomial pp ∈ R[x] with

1 2x 3 p(x)p(x) = 1 2x 3 (x 6 + 1) -(c + c)(x 5 + x) + (c c + c + c)(x 4 + x 2 ) -(c 2 + c 2 + 2) x 3 = T 3 x + x -1 2 -(c + c) T 2 x + x -1 2 + (c c + c + c) T 1 x + x -1 2 - c 2 + c 2 + 2 2 T 0 x + x -1 2 =: g x + x -1 2 .
All the roots of p are contained in T if and only if all the roots of g are contained in [-1, 1], that is, if and only if the solution of system (I) with right hand side

c 1 = -c, c 2 = c, c 3 = -1 is in T 3 .

Characterization via Hermite quadratic forms

Let p, q ∈ R[x] be univariate polynomials with p ̸ = 0. The multiplication by q in the R-algebra R[x]/⟨p⟩ is

m q : R[x]/⟨p⟩ → R[x]/⟨p⟩, f + ⟨p⟩ → q f + ⟨p⟩.
A classical result, usually referred to as Sylvester's version of Sturm's theorem, allows us to effectively characterize when a univariate polynomials has all of its roots in [-1, 1].

Theorem 4.5. All the roots of p are contained in S(q) := {x ∈ R | q(x) ≥ 0} if and only if the Hermite quadratic form

H(p, q) : R[x]/⟨p⟩ → R, f + ⟨p⟩ → Tr(m q f 2 )
is positive semi-definite.

Proof. Let H ∈ R n×n be the symmetric matrix associated to H(p, q) for a fixed basis of R[x]/⟨p⟩, where n = deg(p). Denote by N + , respectively N -, the number of strictly positive, respectively negative, eigenvalues of H, including their multiplicities. By [CLO05, Ch. 2, Thm. 5.2], the rank and the signature of H(p, q) are

N + + N -= Rank(H(p, q)) = |{x ∈ C | p(x) = 0, q(x) ̸ = 0}|, N + -N -= Sign(H(p, q)) = |{x ∈ R | p(x) = 0, q(x) > 0}| =:n+ -|{x ∈ R | p(x) = 0, q(x) < 0}| =:n- .
If all the roots of p are contained in S(q), then n -= 0, and consequently

N + + N -= Rank(H(p, q)) = Sign(H(p, q)) = n + = N + -N -.
This means that N -= 0 and all eigenvalues of H are nonnegative and so H(p, q) is positive semi-definite.

Conversely, assume that H(p, q) is positive semi-definite. Then N -= 0 and so

N + = Sign(H(p, q)) = n + -n -≤ Rank(H(p, q)) = N + ,
Hence, we have n + -n -= Rank(H(p, q)). On the other hand, if not all the roots of p were contained in S(q), then n + -n -would be strictly smaller than Rank(H(p, q)). □

Finally, we compute the matrix of the Hermite quadratric form. If p = x n + c 1 x n-1 + . . . + c n-1 x + c n , then the matrix of the multiplication m x in R[x]/⟨p⟩ in the basis {1, x, . . . , x n-1 } is the companion matrix

      0 0 -c n 1 . . . . . . . . . 0 -c 2 0 1 -c 1       (4.1)
of p, because x x i = x i+1 whenever 0 ≤ i ≤ n -2 and x x n-1 = x n ≡ -c 1 x n-1 -. . . -c n-1 x -c n mod ⟨p⟩.

Corollary 4.6. For c 1 , . . . , c n ∈ R, denote by C ∈ R n×n the matrix from Equation (4.1). Then the solution of system (II) is contained in T n if and only if H ⪰ 0, where H ∈ R n×n has entries

H ij = Tr(C i+j-2 -C i+j ).
Proof. H is the matrix associated to the Hermite quadratic form H(p, q) from Theorem 4.5 with q = 1 -x 2 in the basis {1, x, x 2 , . . . , x n-1 }. Indeed, by [CLO05, Ch. 2, Prop. 4.2], the matrix entries are

H ij = Tr(m q x i-1 x j-1 ) = Tr(m x i+j-2 -x i+j ) = Tr(m i+j-2 x -m i+j x )
for 1 ≤ i, j ≤ n. Since the trace is basis independent, we can replace m x with its matrix C given by Equation (4.1). The statement now follows from Proposition 4.2, because

{x ∈ R | q(x) ≥ 0} = [-1, 1]. □
On the other hand, the univariate Chebyshev polynomials also form a basis

{T 0 , T 1 , . . . , T n-1 } of R[x]/⟨p⟩. If n ≥ 3 and p = T n + d 1 T n-1 + . . . + d n-1 T 1 + d n /2 T 0 ∈ R[x], then the matrix of m x is             0 1/2 0 -d n /4 1 0 . . . -d n-1 /2 1/2 . . . . . . . . . . . . . . . 1/2 -d 3 /2 . . . 0 (1 -d 2 )/2 0 1/2 -d 1 /2             . (4.2)
The entries originate from the recurrence formula 2 x T j = T j+1 + T j-1 . In particular, the last column is obtained through

2 x T n-1 = T n + T n-2 ≡ -d 1 T n-1 + (1 -d 2 ) T n-2 -d 3 T n-3 -. . . -d n-1 T 1 -d n /2 T 0 mod ⟨p⟩.
Corollary 4.7. Let n ≥ 3. For c 1 , . . . , c n-1 ∈ C with c i = (-1) n c n-i and c 0 := (-1) n c n := 1, set

d ℓ = ℓ i=0 c i c ℓ-i ∈ R
for 1 ≤ ℓ ≤ n and denote by C ∈ R n×n the matrix from Equation (4.2). Then the solution of system (I) is contained in T n if and only if H ⪰ 0, where H ∈ R n×n has entries

H ij = Tr(C i+j-2 -C i+j ).
Proof. Using Proposition 4.3, the proof is analogous to the one of Corollary 4.6. □

5 Type A n-1

The group S n acts on R n by permutation of coordinates and leaves the subspace

V := {u ∈ R n | u 1 +. . .+u n = 0} invariant. By [Bou68, Planche I],
A n-1 is a root system in V with fundamental weights

ω i = i j=1 ε j - i n n j=1 ε j = 1 n (n -i, . . . , n -i i times , -i, . . . , -i n-i times ) (1 ≤ i ≤ n -1). ( 5

Orbit polynomials

With W ∼ = S n and Equation (5.1), one observes that a weight µ ∈ Ω is contained in the W-orbit of the first fundamental weight ω 1 if and only if µ ∈ {ω 1 , ω 2 -ω 1 , . . . , ω n-1 -ω n-2 , -ω n-1 }. Hence, the orbit

W • x 1 = {x Be1 | B ∈ W} consists of n distinct monomials in C[x ± ], namely y 1 := x 1 , y 2 := x 2 x -1 1 , . . . , y n-1 := x n-1 x -1 n-2 , y n := x -1 n-1 .
(5.2)

For 1 ≤ i ≤ n, let σ i be the i-th elementary symmetric function in n indeterminates. Recall that, for 1 ≤ i ≤ n -1, the W-invariant orbit polynomial associated to e i ∈ Z n-1 from Equation (3.3) is denoted by

θ i = Θ ei . We show that σ i (y 1 , . . . , y n ) ∈ C[x ± ]
W by expressing it explicitly in terms of the set of fundamental invariants {θ 1 , . . . , θ n-1 }.

Proposition 5.1. In Q[x ± ], we have

σ i (y 1 , . . . , y n ) = n i θ i (1 ≤ i ≤ n -1) and σ n (y 1 , . . . , y n ) = 1.
Proof. It follows from Equation (5.2) that x i = y 1 . . . y i and W acts on the y i by permutation. Hence,

θ i = 1 |W| B∈W x B ei = |Stab W (e i )| |W| J⊆{1,...,n} |J|=i j∈J y j = 1 |W • x i | σ i (y 1 , . . . , y n ).
With |W • x i | = n i and y 1 . . . y n = 1, we obtain the statement. □

The next statement follows from Equation (5.2) and the fact that

x i = y 1 . . . y i for 1 ≤ i ≤ n -1. Lemma 5.2. Let T n 1 := {y ∈ T n | y 1 . . . y n = 1}. The map Υ : T n-1 → T n 1 , x → (y 1 (x), . . . , y n (x))
is bijective.

Hermite characterization with standard monomials

We now characterize, whether a given point z is contained in the T-orbit space T of W, that is, if the equation θ i (x) = z i has a solution x ∈ T n-1 . As shown in previous subsections, this is equivalent to deciding whether a symmetric polynomial system of type (I) has its solution in T n . We state our main result for A n-1 in the standard monomial basis.

Theorem 5.3. For n = 2, we have

T = T = [-1, 1] (type A 1 ). Let n ≥ 3. Define the (n -1)-dimensional R-vector space Z := {z ∈ C n-1 | ∀ 1 ≤ i ≤ n -1 : z i = z n-i } and the matrix polynomial H ∈ Q[z] n×n by H(z) ij = Tr((C(z)) i+j-2 -(C(z)) i+j ), where C(z) =             0 1/2 0 -d n (z)/4 1 0 . . . -d n-1 (z)/2 1/2 . . . . . . . . . . . . . . . 1/2 -d 3 (z)/2 . . . 0 (1 -d 2 (z))/2 0 1/2 -d 1 (z)/2             , d ℓ (z) = (-1) ℓ ℓ i=0 n i n ℓ -i z i z n-ℓ+i z0=zn=1 for 1 ≤ ℓ ≤ n.
For z ∈ Z, we have

H(z) ∈ R n×n and T = {z ∈ Z | H(z) ⪰ 0}. Proof. For z ∈ C n-1 , define c 0 := 1, c n := (-1) n , c i := (-1) i n i z i ∈ C for 1 ≤ i ≤ n -1 as well as d ℓ := d ℓ (z) for 1 ≤ ℓ ≤ n.
To show "⊆", assume that z ∈ T and fix x ∈ T n-1 , such that θ i (x) = z i . By Proposition 5.1 and Lemma 5.2, the unique solution of

(I) σ i (y 1 , . . . , y n ) = (-1) i c i for 1 ≤ i ≤ n is y = Υ(x) ∈ T n
1 . Note that θ j (x) and θ n-j (x) are complex conjugates, because -ω j ∈ W ω n-j . Therefore, z ∈ Z and d ℓ = ℓ i=0 c i c ℓ-i ∈ R yields the last column of C(z). Corollary 4.7 gives us H(z) ⪰ 0. For "⊇" on the other hand, assume z ∈ Z with H(z) ⪰ 0. By Corollary 4.7, the solution y of system (I) is contained in T n 1 . Let x ∈ T n-1 be the unique preimage of y under Υ. Then by Proposition 5.1,

z i = (-1) i n i -1 c i = n i -1 σ i (y 1 , . . . , y n ) = θ i (x) for 1 ≤ i ≤ n -1 and so z = ϑ(x) ∈ T . □ Example: A 2
We investigate the root system from Figure 4 with Weyl group

W ∼ = S 3 . Since W ω 1 = {ω 1 , ω 2 -ω 1 , -ω 2 } and W ω 2 = {ω 2 , ω 1 -ω 2 , -ω 1 }, the integral representation of W is W = 1 0 0 1 , -1 0 1 1 , 1 1 0 -1 , -1 -1 1 0 , 0 1 -1 -1 , 0 -1 -1 0 ⊆ GL 2 (Z).
Let H ∈ Q[z] 3×3 be the matrix polynomial from Theorem 5.3. Then z ∈ Z is contained in T if and only if

H(z) ∈ R 3×3 is positive semi-definite. Let Det(x I 3 -H(z)) = x 3 -h 1 (z) x 2 + h 2 (z) x -h 3 (z) (5.3)
be the characteristic polynomial of H(z). We write z = (z 1 + iz 2 , z 1 -iz 2 ) to simplify the notation. Then z ∈ T ⊆ Z is equivalent to (z 1 , z 2 ) ∈ T ⊆ R 2 and we have

h 3 (z) = -Coeff(x 0 , Det(x I 3 -H(z))) = 2187/64 z 4 2 (3 z 1 + 1) 2 (-3 z 4 1 -6 z 2 1 z 2 2 -3 z 4 2 + 8 z 3 1 -24 z 1 z 2 2 -6 z 2 1 -6 z 2 2 + 1), h 2 (z) = Coeff(x 1 , Det(x I 3 -H(z))) = 243/256 z 2 2 (-243 z 8 1 -972 z 6 1 z 2 2 -1458 z 4 1 z 4 2 -972 z 2 1 z 6 2 -243 z 8 2 + 324 z 7 1 -1620 z 5 1 z 2 2 -4212 z 3 1 z 4 2 -2268 z 1 z 6 2 -432 z 6 1 -2052 z 4 1 z 2 2 -5400 z 2 1 z 4 2 -324 z 6 2 + 180 z 5 1 -3384 z 3 1 z 2 2 -684 z 1 z 4 2 + 18 z 4 1 -804 z 2 1 z 2 2 + 42 z 4 2 + 76 z 3 1 + 404 z 1 z 2 2 -8 z 2 1 -108 z 2 2 + 60 z 1 + 25), h 1 (z) = -Coeff(x 2 , Det(x I 3 -H(z))) = 1/32 (-729 z 6 1 + 1458 z 5 1 + (10935 z 2 2 -1215) z 4 1 + (-2916 z 2 2 + 540) z 3 1 + 351 z 2 2 + 63 +(-10935 z 4 2 + 1458 z 2 2 -135) z 2 1 + (-4374 z 4 2 + 972 z 2 2 + 18) z 1 + 729 z 6 2 -1215 z 4 2 ).
The matrix H(z) is positive semi-definite if and only if h i (z) ≥ 0 for 1 ≤ i ≤ 3. In Figure 5, a solid red line, blue dots and green dashes indicate the varieties of these three polynomials. From those plots, we suspect that the real T-orbit space is invariant under rotation by 2π/3 and horizontal reflection, that is, under the dihedral group D 3 of order 6. The ring of

D 3 -invariants is Q[z 1 , z 2 ] D3 = Q[g 1 , g 2 ] with g 1 := z 2 1 + z 2 2 and g 2 := z 1 (z 2 1 -3 z 2 2 ). One can check that h 3 (z) = 2187/64 z 4 2 (3 z 1 + 1) 2 ≥0 (-6 g 1 (z 1 , z 2 ) -3 g 1 (z 1 , z 2 ) 2 + 8 g 2 (z 1 , z 2 ) + 1) D3-invariant
(5.4) and the variety of the D 3 -invariant factor is the boundary of T .

We now identify the vertices of T , which correspond to the fundamental weights and the origin. With ϑ = θ1+θ2 2 , θ1-θ2 2i , those are

Vertex 0 := ϑ(exp(-2πi ⟨ω 1 , 0⟩), exp(-2πi ⟨ω 2 , 0⟩)) = ϑ(1, 1) = (1, 0) , Vertex 1 := ϑ(exp(-2πi ⟨ω 1 , ω 1 ⟩), exp(-2πi ⟨ω 2 , ω 1 ⟩)) = ϑ exp - 4 3 πi , exp - 2 3 πi = - 1 2 , - √ 3 2 , Vertex 2 := ϑ(exp(-2πi ⟨ω 1 , ω 2 ⟩), exp(-2πi ⟨ω 2 , ω 2 ⟩)) = ϑ exp - 2 3 πi , exp - 4 3 πi = - 1 2 , √ 3 2 .
For a generic point (z 1 , z 2 ) ∈ R 2 , H(z) has rank 3. If h 1 (z) = h 2 (z) = 0, then h 3 (z) = 0 as well and H(z) has rank 0. This holds for two points, namely Vertex 0 and

ϑ(exp(-2πi⟨ω 1 , (ω 1 + ω 2 )/2⟩), exp(-2πi⟨ω 2 , (ω 1 + ω 2 )/2⟩)) = ϑ(exp(-πi), exp(πi)) = - 1 3 , 0 . If h 1 (z) = h 3 (z) = 0 or h 2 (z) = h 3 (z) = 0
, then H(z) has rank 1. This holds for four points, namely Vertex 1 , Vertex 2 and (-1/3, ±2/3). For every other point on the boundary of T , H(z) has rank 2. 6 Type C n

The group {±1} n acts on R n by multiplication of coordinates with ±1. By [Bou68, Planche III], C n is a root system in R n with fundamental weights

ω i = ε 1 + . . . + ε i (1 ≤ i ≤ n). (6.1)
Here, the ε i are the standard basis vectors in R n . The Weyl group is W ∼ = S n ⋉ {±1} n . For 1 ≤ i ≤ n, we have -ω i ∈ W ω i (because -I n ∈ W) and the orbit W ω i has cardinality 2 i n i . Since the rank is n, the ring of Laurent polynomials is

C[x ± ] = C[x 1 , x -1 1 , . . . , x n , x -1 n ] and the polynomial ring is Q[z] = Q[z 1 , . . . , z n ].

Orbit polynomials

By Equation (6.1), the orbit

W • x 1 = {x Be1 | B ∈ W} consists of 2n distinct monomials in C[x ± ],
namely

y 1 := x 1 , y 2 := x 2 x -1 1 , . . . , y n := x n x -1 n-1 (6.2)
and their inverses. For 1 ≤ i ≤ n, let σ i be the i-th elementary symmetric function in n indeterminates and recall that θ i = Θ ei is the W-invariant orbit polynomial associated to e i ∈ Z n from Equation (3.3).

Proposition 6.1. In C[x ± ], we have

σ i y 1 + y -1 1 2 , . . . , y n + y -1 n 2 = n i θ i (1 ≤ i ≤ n).
Proof. It follows from Equation (6.2) that x i = y 1 . . . y i and W acts on the y ±1 i by permutation and inversion. Hence,

θ i = 1 |W| B∈W x B ei = |Stab W (e i )| |W| J⊆{1,...,n} |J|=i δ∈{±1} J j∈J y δj j = 1 |W • x i | J⊆{1,...,n} |J|=i j∈J (y j + y -1 j ) With |W • x i | = 2 i n i
, we obtain the statement. □

The next statement follows from Equation (6.2) and the fact that

x i = y 1 . . . y i for 1 ≤ i ≤ n.
Lemma 6.2. The map Υ :

T n → T n , x → (y 1 (x), . . . , y n (x)),
is bijective.

Hermite characterization with standard monomials

We now characterize, whether a given point z is contained in the T-orbit space T of W, that is, if the equation θ i (x) = z i has a solution x ∈ T n . As shown in previous subsections, this is equivalent to deciding whether a symmetric polynomial system of type (II) has its solution in T n . We state our main result for C n in the standard monomial basis.

Theorem 6.3. Define the matrix polynomial

H ∈ Q[z] n×n by H(z) ij = Tr(C(z) i+j-2 -C(z) i+j ), where C(z) =      0 • • • 0 -c n (z) 1 0 -c n-1 (z) . . . . . . 0 1 -c 1 (z)      , c i (z) = (-1) i n i z i for 1 ≤ i ≤ n. Then T = T = {z ∈ R n | H(z) ⪰ 0}. Proof. Let z ∈ R n and set c i := c i (z) ∈ R for 1 ≤ i ≤ n.
To show "⊆", assume that z ∈ T . Then there exists x ∈ T n , such that θ i (x) = z i for 1 ≤ i ≤ n. By Proposition 6.1 and Lemma 6.2, the solution of the symmetric polynomial system

(II) σ i y 1 + y -1 1 2 , . . . , y n + y -1 n 2 = (-1) i c i (1 ≤ i ≤ n) is y = Υ(x) ∈ T n . Applying Corollary 4.6 yields H(z) ⪰ 0.
For "⊇" on the other hand, assume H(z) ⪰ 0. By Corollary 4.6, the solution y of the above system (II) is contained in T n . Let x ∈ T n be the unique preimage of y under Υ. Then

z i = θ i (x) and so z = ϑ(x) is contained in T . □ Example: C 2
We study the root system from Figure 6 with Weyl group

W ∼ = S 2 ⋉{±1} 2 . Since W ω 1 = {±ω 1 , ±(ω 1 -ω 2 )} and W ω 2 = {±ω 2 , ±(2ω 1 -ω 2 )}, the integral representation of W is W = 1 0 0 1 , -1 0 1 1 , 1 2 0 -1 , 1 2 -1 -1 , -1 -2 1 1 , -1 -2 0 1 , 1 0 -1 -1 , -1 0 0 -1 ⊆ GL 2 (Z).
Then z = (z 1 , z 2 ) ∈ R 2 is contained in T if and only if the resulting Hermite matrix

H(z) = -4 z 2 1 + 2 z 2 + 2 -8 z 3 1 + 6 z 1 z 2 + 2 z 1 -8 z 3 1 + 6 z 1 z 2 + 2 z 1 -16 z 4 1 + 16 z 2 1 z 2 + 4 z 2 1 -2 z 2 2 -2 z 2 . (6.3)
is positive semi-definite, or equivalently, its determinant and trace

Det(H(z)) = -4 (z 2 1 -z 2 ) (2 z 1 + 1 + z 2 ) (2 z 1 -1 -z 2 ) Tr(H(z)) = -16 z 4 1 + 16 z 2 1 z 2 -2 z 2 2 + 2 (6.4)
are nonnegative. The varieties of these two polynomials are depicted in Figure 7. If z ∈ R 2 with Det(H(z)) = Tr(H(z)) = 0, then H(z) has rank 0 and z must be one of the three vertices Vertex 0 := ϑ(exp(-2πi ⟨ω 1 , 0⟩), exp(-2πi ⟨ω 2 , 0⟩)) = ϑ(1, 1) = (1, 1), Vertex 1 := ϑ(exp(-2πi ⟨ω 1 , ω 1 /2⟩), exp(-2πi ⟨ω 2 , ω 1 /2⟩)) = ϑ(-1, -1) = (0, -1),

Vertex 2 := ϑ(exp(-2πi ⟨ω 1 , ω 2 /2⟩), exp(-2πi ⟨ω 2 , ω 2 /2⟩)) = ϑ(-1, 1) = (-1, 1).
The boundary of T is dictated by the determinant, but from the positivity condition one can observe that the trace is also required. Alternatively, the inequality given by the trace could be replaced by the constraint that the orbit space is contained in the square [-1, 1] 2 , which is always true. Hence, H(z) has rank 1 if z is a boundary point, but not a vertex, and rank 2 if z is generic. 7 Type B n

The root system B n from [Bou68, Planche II] is a root system in R n . Its Weyl group is the same as that of C n , that is W ∼ = S n ⋉ {±1} n , but now the fundamental weights are Since the rank is n, the ring of Laurent polynomials is

ω i = ε 1 + . . . + ε i (1 ≤ i ≤ n -
C[x ± ] = C[x 1 , x -1 1 , . . . , x n , x -1 n ] and the polynomial ring is Q[z] = Q[z 1 , . . . , z n ].

Orbit polynomials

By Equation (7.1), the orbit W

• x 1 = {x Be1 | B ∈ W} consists of 2 n distinct monomials in C[x ± ], namely y 1 := x 1 , y 2 := x 2 x -1 1 , . . . , y n-1 := x n-1 x -1 n-2 , y n := x 2 n x -1 n-1 (7.2)
and their inverses. For 1 ≤ i ≤ n, let σ i be the i-th elementary symmetric function in n indeterminates and recall that θ i = Θ ei is the W-invariant orbit polynomial associated to e i ∈ Z n from Equation (3.3).

Proposition 7.1. In C[x ± ], we have

σ i y 1 + y -1 1 2 , . . . , y n + y -1 n 2 = n i θ i (1 ≤ i ≤ n -1)
and

σ n y 1 + y -1 1 2 , . . . , y n + y -1 n 2 = Θ 2 en .
Proof. By Equation (7.2), x i = y 1 . . . y i and x 2 n = y 1 . . . y n . The proof is analogous to Proposition 6.1. □

We compute the missing explicit expression for the right hand side of Proposition 7.1 in terms of the fundamental invariants θ i .

Lemma 7.2. In C[x ± ], we have

Θ 2 en = 2 n θ 2 n - n-1 j=1 n j θ j -1.
Proof. The cardinality of the orbit W e n ⊆ Z n is 2 n . Let α ∈ W e n and distinguish as follows.

1. If α = e n , then Θ en+α = Θ 2 en is the term for which we seek an explicit formula.

2. If α = -e n , then Θ en+α = Θ 0 = 1.

3. Otherwise, it follows from Equation (7.1) that there exists 1 ≤ j ≤ n -1, such that e n + α ∈ W e j . The number of α, for which this is the case, is n j .

With the recurrence formula from Equation (3.4), we conclude

2 n θ 2 n = |W e n | Θ en Θ en = α∈W en Θ α+en = Θ 2 en + n-1 j=1 n j Θ ej + 1 = Θ 2 en + n-1 j=1 n j θ j + 1
and obtain the formula for Θ 2 en . □ Lemma 7.3. The map Υ : T n → T n , x → (y 1 (x), . . . , y n (x)), is surjective. Furthermore, every y ∈ T n has exactly two distinct preimages x, x ′ ∈ T n with

θ i (x) = θ i (x ′ ) (1 ≤ i ≤ n -1) and θ n (x) = -θ n (x ′ ).
Proof. For y ∈ T n , choose x ∈ T n with x 1 = y 1 , x 2 = y 1 x 1 , . . . , x n-1 = y n-1 x n-2 and x 2 n = y n x n-1 . Then x is a preimage of y under Υ and unique up to a sign in the last coordinate.

For 1 ≤ i ≤ n -1, we have θ i (x) = θ i (x ′ ), because y k (x) = y k (x ′ ) for all 1 ≤ k ≤ n.

It follows from Equation (7.1) that the stabilizer of ω n in W ∼ = S n ⋉ {±1} n is isomorphic to the subgroup S n . Hence, we have W ω n = {±1} n ω n and, for all µ ∈ W ω n , there exist ϵ i = ±1 and ν ∈ Z ω 1 ⊕. . .⊕Z ω n-1 , such that

µ = ϵ 1 2 ε 1 + . . . + ϵ n 2 ε n = ϵ 1 2 ω 1 + n-1 i=2 ϵ i 2 (ω i -ω i-1 ) + ϵ n 2 (2 ω n -ω n-1 ) = ϵ n ω n + ν.
Now let α, β ∈ Z n , such that µ = α 1 ω 1 + . . . + α n ω n and ν = β 1 ω 1 + . . . + β n ω n . Then β n = 0 and the monomial in θ n corresponding to µ is x α = x β1 1 . . . x βn-1 n-1 x ϵn n . Thus, x α is linear in x n . Since every monomial in θ n can be written in terms of such β and ϵ i , the polynomial θ n is linear in x n and, with x, x ′ as above, we have θ n (x) = -θ n (x ′ ). □

Hermite characterization with standard monomials

We now characterize, whether a given point z is contained in the T-orbit space T of W, that is, if the equation θ i (x) = z i has a solution x ∈ T n . As shown in previous subsections, this is equivalent to deciding whether a symmetric polynomial system of type (II) has its solution in T n . We state our main result for B n in the standard monomial basis.

Theorem 7.4. Define the matrix polynomial

H ∈ Q[z] n×n by H(z) ij = Tr(C(z) i+j-2 -C(z) i+j ), where C(z) =      0 • • • 0 -c n (z) 1 0 -c n-1 (z) . . . . . . 0 1 -c 1 (z)      , c i (z) = (-1) i n i z i (1 ≤ i ≤ n -1) and c n (z) = (-1) n 2 n z 2 n - n-1 i=1 n i z i -1 . Then T = T = {z ∈ R n | H(z) ⪰ 0}. Proof. Let z ∈ R n and set c i := c i (z) ∈ R for 1 ≤ i ≤ n.
To show "⊆", assume that z ∈ T . Then there exists x ∈ T n , such that θ i (x) = z i for 1 ≤ i ≤ n. By Proposition 7.1 and Lemmas 7.2 and 7.3, the solution of the symmetric polynomial system

(II) σ i y 1 + y -1 1 2 , . . . , y n + y -1 n 2 = (-1) i c i for 1 ≤ i ≤ n is y = Υ(x) ∈ T n . Applying Corollary 4.6 yields H(z) ⪰ 0.
For "⊇" on the other hand, assume H(z) ⪰ 0. By Corollary 4.6, the solution y of system (II) with coefficients c i is contained in T n . According to Lemma 7.3, y has exactly two distinct preimages x, x ′ ∈ T n under Υ with

x 1 = x ′ 1 , . . . , x n-1 = x ′ n-1 and x n = -x ′ n . We have z i = θ i (x) = θ i (x ′ ) for 1 ≤ i ≤ n -1 and z 2 n = θ n (x) 2 = θ n (x ′ ) 2 with θ n (x) = -θ n (x ′ ). Therefore, z n = θ n (x) or z n = -θ n (x) = θ n (x ′ ) and thus, z is contained in T . □ 1. If n is even, then Θ en-1+en = 2 n-1 n θ n-1 θ n - 1 n (n-2)/2 j=1 n 2j -1 θ 2j-1 , Θ 2 en-1 = 2 n-1 θ 2 n-1 - (n-2)/2 j=1 n 2j θ 2j -1, Θ 2 en = 2 n-1 θ 2 n - (n-2)/2 j=1 n 2j θ 2j -1. 2. If n is odd, then Θ en-1+en = 2 n-1 n θ n-1 θ n - 1 n (n-3)/2 j=1 n 2j θ 2j - 1 n , Θ 2 en-1 = 2 n-1 θ 2 n-1 - (n-3)/2 j=0 n 2j + 1 θ 2j+1 , Θ 2 en = 2 n-1 θ 2 n - (n-3)/2 j=0 n 2j + 1 θ 2j+1 .
Proof. With Equation (8.1) and the recurrence formula from Equation (3.4), the proof is analogous to that for the case of B n in Lemma 7.2. A rigorous computation can be found in [Met22, Lemma 1.43]. □ Lemma 8.3. The map Υ : T n → T n , x → (y 1 (x), . . . , y n (x)), is surjective. Furthermore, every y ∈ T n has exactly two distinct preimages x, x ′ ∈ T n under Υ with

θ i (x) = θ i (x ′ ), if 1 ≤ i ≤ n -2 -θ i (x ′ ), if i ∈ {n -1, n} . Proof. For y ∈ T n , choose x ∈ T n with x 1 = y 1 , x 2 = y 1 x 1 , . . . , x n-2 = y n-2 x n-3 and x 2 n-1 = y -1 n y n-1 x n-2 , x n = y n x n-1 .
Then x is a preimage of y under Υ and unique up to a common sign in x n-1 and x n . This second preimage is denoted by x ′ . By Proposition 8.1, we have θ

i (x) = θ i (x ′ ) whenever 1 ≤ i ≤ n -2.
It follows from Equation (8.1) that the stabilizers of ω n-1 and ω n in W are both S n . Hence, we have

W ω n-1 = {±1} n-1 ω n-1 and W ω n = {±1} n-1 ω n . Thus, for µ ∈ W ω n , there exist ϵ i = ±1 with ϵ 1 . . . ϵ n = 1 and ν ∈ Z ω 1 ⊕ . . . ⊕ Z ω n-2 , such that µ = ϵ 1 2 ε 1 + . . . + ϵ n 2 ε n = ϵ 1 2 ω 1 + n-2 i=2 ϵ i 2 (ω i -ω i-1 ) + ϵ n-1 2 (ω n + ω n-1 -ω n-2 ) + ϵ n 2 (ω n -ω n-1 ) = ϵ n-1 + ϵ n 2 ω n + ϵ n-1 -ϵ n 2 ω n-1 + ν ∈ Ω.
Now let α, β ∈ Z n , such that µ = α 1 ω 1 + . . . + α n ω n and ν = β 1 ω 1 + . . . + β n ω n . Then β n-1 = β n = 0 and the monomial in θ n corresponding to µ is

x α = x β1 1 . . . x βn-2 n-2 x (ϵn-1-ϵn)/2 n-1 x (ϵn-1+ϵn)/2 n z 2 n-1 + z 2 n = θ n-1 (x) 2 + θ n (x) 2 = θ n-1 (x ′ ) 2 + θ n (x ′ ) 2 and z n-1 z n = θ n-1 (x) θ n (x) = θ n-1 (x ′ ) θ n (x ′ ). Therefore, {z n-1 , z n } ∈ {{θ n-1 (x), θ n (x)}, {θ n-1 (x ′ ), θ n (x ′ )}}. If z n-1 = θ n-1 (x), z n = θ n (x), then z = ϑ(x).
Otherwise, set x = (x 1 , . . . , x n-2 , x n , x n-1 ) to obtain z n-1 = θ n-1 (x), z n = θ n (x) and z = ϑ(x). An analogous argument applies to x ′ and thus, z is contained in T . □

9 Type G 2

The group S 3 ⋉ {±1} acts on R 3 by permutation of coordinates and scalar multiplication with ±1 and leaves the subspace

V = {u ∈ R 3 | u 1 + u 2 + u 3 = 0} invariant. By [Bou68, Planche IX],
G 2 is a root system in V with rank 2 and fundamental weights

ω 1 = [1, -1, 0] t and ω 2 = [-2, 1, 1] t . (9.1) The Weyl group of G 2 is W ∼ = S 3 ⋉ {±1}. We have -ω 1 ∈ W ω 1 and -ω 2 ∈ W ω 2 . Furthermore, |W ω 1 | = |W ω 2 | = 6, see Figure 10.
Since the rank is 2, the ring of Laurent polynomials is

C[x ± ] = C[x 1 , x -1 1 , x 2 , x -1 2 ] and the polynomial ring is Q[z] = Q[z 1 , z 2 ].

Orbit polynomials

The integral representation of W is

W = ± 1 0 0 1 , ± -1 0 1 1 , ± 1 3 0 -1 , ± 1 3 -1 -2 , ± 2 3 -1 -1 , ± 2 3 -1 -2 ⊆ GL 2 (Z).
Then the orbit W

• x 1 = {x Be1 | B ∈ W} ⊆ C[x ± ] consists of 6 distinct monomials, namely y 1 := x 1 , y 2 := x 1 x -1 2 , y 3 := x -2 1 x 2 (9.2)
and their inverses. Let σ 1 , σ 2 , σ 3 be the elementary symmetric functions in 3 indeterminates and θ 1 = Θ e1 , θ 2 = Θ e2 be the W-invariant orbit polynomials associated to e 1 , e 2 ∈ Z 2 from Equation (3.3).

and shall have characteristic polynomial Lemma 10.3. For x ∈ T r and y ±1 1 , . . . , y ±1 n monomials as in Lemma 10.2, let C ∈ R n×n be a matrix with eigenvalues (y i (x) + y i (x) -1 )/2. Then, for k ∈ N, the trace of the k-th power of C is For x ∈ T r and z := ϑ(x) ∈ T ⊆ Z, it is now essential to note that C(z) ∈ R n×n has eigenvalues (y i (x) + y i (x) -1 )/2 ∈ [-1, 1] with y ±1 1 , . . . , y ±1 n as in Lemma 10.2. This follows from the construction, by which C is the matrix of a multiplication map, see Section 4.2. In particular, we obtain a formula for the entries H(z) ij with Lemma 10.3. In the basis of (real) generalized Chebyshev polynomials from Definition 3.5 and Proposition 3.8, we have Θ (k-2 ℓ) e1 (x) + Θ (2 ℓ-k) e1 (x) = T (k-2 ℓ) e1 (ϑ(x)) + T (2 ℓ-k) e1 (ϑ(x)) = 2 T (k-2 ℓ) e1 ( ϑ(x)).

Det(x I 3 -H(z)) = x 3 -h 1 (z) x 2 + h 2 (z) x -h 3 (z). ( 9
We now substitute z → z := ϑ(x) ∈ T ⊆ R n . For 1 ≤ i, j ≤ n and k := i + j, the entry H(z) ij is

Tr(C(z) k-2 ) -Tr(C(z) k ) = 2 n 2 k   4 ⌈k/2⌉-2 ℓ=0 k -2 ℓ T (k-2 (ℓ+1)) e1 ( z) - ⌈k/2⌉-1 ℓ=0 k ℓ T (k-2 ℓ) e1 ( z)   + n 2 k 4 k-2 k/2-1 -k k/2 , k even 0, k odd = 2 n 2 k   4 ⌈k/2⌉-1 ℓ=1 k -2 ℓ -1 T (k-2 ℓ) e1 ( z) - ⌈k/2⌉-1 ℓ=0 k ℓ T (k-2 ℓ) e1 ( z)   + n 2 k 4 k-2 k/2-1 -k k/2 , k even 0, k odd = 2 n 2 k   -T k e1 ( z) + ⌈k/2⌉-1 ℓ=1 4 k -2 ℓ -1 - k ℓ T (k-2 ℓ) e1 ( z)   + n 2 k 4 k-2 k/2-1 -k k/2 , k even 0, k odd .
Dividing by 2 n does not change whether H( z) is positive semi-definite and so we obtain the formula. □ 2. H has Hankel structure, that is, H ij = H kℓ whenever i + j = k + ℓ. Therefore, the number of distinct entries is at most 2 n -1.

3. Denote by H (n) the matrix H for fixed n. Then the leading principal submatrices of H have the structure of H (k) for k ≤ n.

Conclusion and future work

We have characterized the T-orbit space of a Weyl group acting nonlinearly on the compact torus as a compact basic semi-algebraic set. The characterization is given through a polynomial matrix inequality and holds for the Weyl groups associated to root systems of type A n-1 , B n , C n , D n and G 2 . It is the first explicit characterization for nonlinear actions in the sense that we have given a closed formula for the matrix. By expressing the entries in the basis of generalized Chebyshev polynomials, we have shown that these matrices have a common structure.

We plan to address the remaining cases E 6 , E 7 , E 8 and F 4 in a future work.

Figure 2 :

 2 Figure 2: The graphs of f (left) and g with the new feasible region (right), see Example 2.1.

Figure 3 :

 3 Figure 3: The root system C 2 admits a quadratic weight lattice (middle). Note that the change of basis ω i → e i is not the identity (left). The circles • are the weights and the green dots • are the roots. The blue dots • and red dots • are the orbits of the fundamental weights and • is the origin. The T-orbit space is indicated by a dashed line (right).

  .1) Here, the ε i are the standard basis vectors in R n . The weight lattice is Ω := {u/n | u ∈ V ∩ Z n } and the Weyl group is W ∼ = S n . For 1 ≤ i ≤ n -1, we have -ω i ∈ W ω n-i and the orbit W ω i has cardinality n i . Since the rank is n -1, the ring of Laurent polynomials is C[x ± ] = C[x 1 , x -1 1 , . . . , x n-1 , x -1 n-1 ] and the polynomial ring is Q[z] = Q[z 1 , . . . , z n-1 ].

Figure 4 :

 4 Figure 4: The root system A 2 and its weight lattice. The orbits of the fundamental weights are the blue and red lattice elements.

Figure 5 :

 5 Figure 5: The intersection of the semi-algebraic sets defined by the coefficients of the characteristic polynomial of H(z) is the T-orbit space associated to A 2 .

Figure 6 :

 6 Figure 6: The root system C 2 and its weight lattice. The orbits of the fundamental weights are the blue and red lattice elements.

Figure 7 :

 7 Figure 7: The semi-algebraic sets defined by the determinant and trace of H(z).

Figure 8 :

 8 Figure 8: The root system B 2 and its weight lattice. The orbits of the fundamental weights are the blue and red lattice elements.

Figure 10 :

 10 Figure 10: The root system G 2 and the weight lattice. The orbits of the fundamental weights are the blue and red lattice elements.

  .3) Again, we have z ∈ T if and only if h i (z) ≥ 0 for 1 ≤ i ≤ 3 withh 3 (z) = -Coeff(x 0 , Det(x I 3 -H(z)))= 81/16 (24z 3 1 -12 z 1 z 2 -z 2 2 -6 z 1 -4 z 2 -1) (3 z 2 1 -2 z 2 -1) 2 (3 z 1 + 1) 2 , h 2 (z) = Coeff(x 1 , Det(x I 3 -H(z))) = 27/16 (3 z 2 1 -2 z 2 -1) (648 z 6 1 -540 z 5 1 -972 z 4 1 z 2 -216 z 3 1 z 2 2 -648 z 4 1 + 36 z 3 1 z 2 +369 z 2 1 z 2 2 + 126 z 1 z 3 2 + 9 z 4 2 -18 z 3 1 + 480 z 2 1 z 2 + 246 z 1 z 2 2 + 36 z 3 2 + 129 z 2 1 + 202 z 1 z 2 +37 z 2 2 + 44 z 1 + 28 z 2 + 4), h 1 (z) = -Coeff(x 2 , Det(x I 3 -H(z))) = 9/4 -729 z 6 1 + 243 z 4 1 + 729 z 4 1 z 2 -729/4 z 2 1 z 2 2 + 243/2 z 3 1 z 2 + 189/4 z 3 1 -405/4 z 2 1 z 2 -243/4 z 1 z 22 + 27/4 z 3 2 -81/4 z 2 1 -81/2 z 1 z 2 -27/4 z 2 2 -9/2 z 1 -9/2 z 2 .

Figure 11 :

 11 Figure 11: The semi-algebraic sets defined by the coefficients of the characteristic polynomial of H(z).

  k-2 ℓ) e1 (x) + Θ (2 ℓ-k) e1 (x)The trace of the k-th power of C is the sum of the k-th powers of the eigenvalues. Thus,Tr(C k ) = n i=1 y i (x) + y i (x) x) k-ℓ y i (x) x) k-2 ℓ + y i (x) 2 ℓ-k + n 2 k k k/2 , k even 0, k odd .Applying Lemma 10.2 yields the formula. □ Proof of Theorem 10.1. The T-orbit space of W is by definition T = {ϑ(x) | x ∈ T r }. According to Theorems 5.3, 6.3, 7.4, 8.4 and 9.3, we have T = {z ∈ Z | H(z) ⪰ 0}. Here, Z is an R-vector space with dim(Z) = r and H ∈ Q[z] n×n is a matrix with entries H ij = Tr(C i+j-2 ) -Tr(C i+j ), where C ∈ Q[z] n×n is defined in the respective theorem.

Remark 10. 4 .Te 1 -T3 e 1 8 T0-T4 e 1 16 2 8 T0-T4 e 1 16 2T0-T4 e 1 16 2525

 4181681616 The matrix polynomial H ∈ Q[ z] n×n from Theorem 10.1 follows the pattern Te 1 -T3 e 1 -T5 e 1 32• • •Te 1 -T3 e 1 Te 1 -T3 e 1 -T5 e 1 32 2 T0+ T2 e 1 -2 T4 e 1 -T6 e 1 64• • • Te 1 -T3 e 1 -T5 e 1 32 2 T0+ T2 e 1 -2 T4 e 1 -T6 e 164 Te 1 -T3 e 1 -3 T5 e 1 -T7 e 1 128• • • Te 1 -T3 e 1 -T5 e 1 32 2 T0+ T2 e 1 -2 T4 e 1 -T6 e 164 Te 1 -T3 e 1 -3 T5 e 1 -T7 e 1 128 5 T0+4 T2 e 1 -4 T4 e 1 -4 T6 e 1 -T8 e 1 256 only T α occurring in H are those, where α ∈ N r is a multiple of e 1 . An entry is a linear combination of at most n + 1 such polynomials. The largest possible degree of an entry is 2 n = deg( T 2 n e1 ).

  The distinct Θ α form a basis for C[x ± ] W as a C-vector space [Lor05, Eq. (3.4)]. ± ] W is finitely generated [Lor05, Coro. 3.3.2]. A minimal set of generators is called a set of fundamental invariants for C[x ± ] W . Let {θ 1 , . . . , θ m } ⊆ C[x ± ] W be a set of fundamental invariants. The map

	As a C-algebra, C[x Proposition 3.1.
	We have Θ 0 = 1 and, for α, β ∈ Z n ,	Θ α Θ β =	1 |W|	B∈W	Θ α+Bβ .	(3.4)

https://github.com/TobiasMetzlaff/GeneralizedChebyshev.

Acknowledgments

The authors wish to acknowledge the insights that Claudio Procesi (Rome) shared and are grateful for discussions with Philippe Moustrou (Toulouse) and Robin Schabert (Tromsø).

The majority of the work of Tobias Metzlaff was carried out during his doctoral studies [Met22] at Inria d'Université Côte d'Azur, supported by European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie Actions, grant agreement 813211 (POEMA). Minor changes were applied during his postdoctoral research at RPTU Kaiserslautern-Landau, supported by the Deutsche Forschungsgemeinschaft transregional collaborative research centre (SFB-TRR) 195 "Symbolic Tools in Mathematics and their Application".

Example: B 2

We study the root system from Figure 8 with Weyl group W ∼ = S 2 ⋉ {±1} 2 . Since Wω 1 = {ω 1 , -ω 1 , ω 1 -2ω 2 , 2ω 2 -ω 1 } and Wω 2 = {ω 2 , -ω 2 , ω 1 -ω 1 , ω 2 -ω 1 }, the integral representation of W is

Then z = (z 1 , z 2 ) ∈ R 2 is contained in T if and only if the resulting Hermite matrix

is positive semi-definite, or equivalently, its determinant and trace Det(H(z)) = -64 z 2 2 (-z 2 2 + z 1 ) (z 1 + 1 + 2 z 2 ) (z 1 + 1 -2 z 2 ) Tr(H(z)) = -16 z 4 1 + 64 z 2 1 z 2 2 -32 z 4 2 -32 z 3 1 + 32 z 1 z 2 2 -24 z 2 1 + 16 z 2 2 -8 z 1 (7.4) are nonnegative. The varieties of these two polynomials are depicted below. Let us confirm that it follows also from our semi-algebraic description: Indeed, we have Det(H(z))/Det(H C2 (z 2 , z 1 )) = 16 z 2 2 ≥ 0 on R 2 and the variety of the determinant dictates the boundary. On the other hand, the determinant of H(z) vanishes if z 2 = 0 and the rank drops to 0 in z = (0, 0) although this is not a vertex. The other points which correspond to H having rank 0 are precisely the three vertices

Hence, H(z) has rank 1 if z is a boundary point, but not a vertex or 0, and rank 2 if z is generic.

Type D n

The group {±1} n-1 acts on R n by scalar multiplication of coordinates with ±1, where only an even amount of sign changes is admissible. By [Bou68, Planche IV], D n is a root system in R n with fundamental weights

Here, the ε i are the standard basis vectors in R n . The Weyl group of D n is W ∼ = S n ⋉ {±1} n-1 . For all 1 ≤ i ≤ n, we have -ω i ∈ W ω i , unless n is odd. In this case, -ω n-1 ∈ W ω n . Furthermore, we have

Orbit polynomials

By Equation (8.1), the orbit

and their inverses. For 1 ≤ i ≤ n, let σ i be the i-th elementary symmetric function in n indeterminates and recall that θ i = Θ ei is the W-invariant orbit polynomial associated to e i ∈ Z n from Equation (3.3).

Proposition 8.1. In C[x ± ], we have

It follows from Equation (8.2) that x i = y 1 . . . y i . Then the statement for θ i is proven analogously to Proposition 6.1.

With x n x n-1 = y 1 . . . y n-1 , we obtain the equation for Θ en-1+en as well.

Finally, we have

where y ϵ = y ϵ1 1 . . . y ϵn n . This proves the last equation. □

We give the explicit expression for the right hand side of Proposition 8.1 in terms of the fundamental invariants.

Lemma 8.2. In C[x ± ], the following statements hold.

with (ϵ n-1 ± ϵ n )/2 ∈ {-1, 0, 1}. Therefore, x α is linear in x n-1 and independent of x n or vice versa. With x, x ′ as above we have x α = -(x ′ ) α . Since every monomial in θ n can be written in terms of such β and ϵ i , we obtain θ n (x) = -θ n (x ′ ). Analogously, for µ ∈ W ω n-1 , we have ϵ 1 . . . ϵ n = -1 and obtain the statement for θ n-1 . □

Hermite characterization with standard monomials

We now characterize, whether a given point z is contained in the T-orbit space T of W, that is, if the equation θ i (x) = z i has a solution x ∈ T n . As shown in previous subsections, this is equivalent to deciding whether a symmetric polynomial system of type (II) has its solution in T n . We state our main result for D n in the standard monomial basis.

Theorem 8.4. Define the n-dimensional R-vector space

if n is odd and the matrix polynomial

To show "⊆", assume that z ∈ T . Then there exists x ∈ T n , such that θ i (x) = z i for 1 ≤ i ≤ n. Furthermore, we have z ∈ Z and c i ∈ R. By Proposition 8.1 and Lemmas 8.2 and 8.3, the solution of the symmetric polynomial system

For "⊇" on the other hand, assume z ∈ Z with H(z) ⪰ 0. Hence, c i ∈ R and by Corollary 4.6, the solution y of the above system (II) is contained in T n . According to Lemma 8.3, y has two distinct preimages x, x ′ ∈ T n . We have

Proof. In principle, this can be checked by hand, but we still give a recipe to obtain the equations: It follows from Equation (9.2) that the product y 1 y 2 y 3 is 1 and moreover

2 = y 2 2 y 3 . Note that we do not need the inverses of the y i . Thus, after computing the orbits W • x 1 and W • x 2 , we find polynomials g 1 , g 2 ∈ R[y 1 , y 2 , y 3 ] , such that θ i = g i (y). Consider the ideal

Choose an elimination ordering with {y 1 , y 2 , y 3 } ⪰ {θ 1 , θ 2 } and let G be a Gröbner basis of I. Note that the expression on the left-hand side of the statement is W-invariant. We can write it as σ i (y 1 + y 2 y 3 , y 2 + y 1 y 3 , y 3 + y 1 y 2 )/2 i , that is, a polynomial in Q[y 1 , y 2 , y 3 , θ 1 , θ 2 ]. Because of the W-invariance, the normal form of this polynomial with respect to G only depends on θ 1 and θ 2 . One obtains the right-hand side. □

The next statement follows immediately from Equation (9.2).

Lemma 9.2. The map Υ :

is bijective.

Hermite characterization with standard monomials

With Proposition 9.1 and Lemma 9.2, the proof of the following is analogous to Theorem 6.3.

Example

Our result applies to the root system from Figure 10 as follows. Let z = (z 1 , z 2 ) ∈ R 2 . The matrix polynomial from Theorem 9.3 evaluated in z is 4/3

The vertices are

Vertex 3 := ϑ(exp(-2πi ⟨ω 1 , 0⟩), exp(-2πi ⟨ω 2 , 0⟩)) = ϑ(1, 1) = (1, 1).

For a generic point (z 1 , z 2 ) ∈ R 2 , H(z) has rank 3. If h 1 (z) = h 2 (z) = 0, then h 3 (z) = 0 as well and H(z) has rank 0. This holds for two points, namely Vertex 0 and

If h 1 (z) = h 3 (z) = 0 or h 2 (z) = h 3 (z) = 0, then H(z) has rank 1. This holds for four points, namely Vertex 1 , Vertex 2 and (-1/3, ±2/3). For every other point on the boundary of T R , H(z) has rank 2.

10 Hermite characterization of T R with Chebyshev polynomials

Finally, we give a unifying formula for our constructed Hermite matrices in terms of the real generalized Chebyshev polynomials T α from Proposition 3.8. Those depend on the root systems and so the matrices are not identical in Q[z]. Instead, unifying means that the formulae follow the same pattern in the Chebyshev basis.

We have to distinguish carefully between the size of the Hermite matrix and the number of coordinates of the T-orbit space: Throughout, let n ∈ N and R be a root system of type A n-1 , B n , C n , D n or G 2 . In particular, the rank of R is r := n -1 for A n-1 , r := 2 for G 2 and r := n otherwise. A common property of these types is that the symmetric group S n is a semi-direct factor of the Weyl group. This fact made the characterization via Hermite quadratic forms in the previous Sections 5 to 9 possible. The r-variate polynomial ring is denoted by Q[z] = Q[z 1 , . . . , z r ], respectively Q[ z] = Q[ z 1 , . . . , z r ].

Theorem 10.1. Let H ∈ Q[ z] n×n be the matrix polynomial with entries

Then the real T-orbit space of the multiplicative action of W on

We need two lemmata involving the orbit polynomials Θ α = 1 |W| B∈W

x Bα for α ∈ Z r from Equation (3.3).

Lemma 10.2. The set W • x 1 ∪ W • x -1 1 consists of n distinct monomials and their inverses, which are denoted by y ±1 1 , . . . , y ±1 n . For x ∈ T r and k ∈ N, we have (y i (x) k + y i (x) -k )/2 ∈ [-1, 1] and

Proof. The y i are the monomials from Equations (5.2) to (9.2). The left hand side is 1/n σ 1 evaluated in y i (x) k + y i (x) -k . Then the proof is analogous to Propositions 5.1, 6.1, 7.1, 8.1 and 9.1. □


Emmanuel Fara

Jean-David Moreau

Léo Szewczyk

Georges Gand

Laurent Mendras

Rolande Fayolle

Nicolas Klee

Suzanne Jiquel

Rémi Laffont

Lauriane Poloni

Leo Szewczyk

Michel Verolet

Emmanuelle Vennin

Étude des traces de pas de vertébrés du Permien et du Trias d'Ardèche. Enjeux et perspectives

Keywords: Empreintes de pas fossiles, Reptiles, Permien, Trias, Géoparc mondial UNESCO

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

Introduction

Les périodes géologiques du Permien (299-252 millions d'années) et du Trias (252-201 millions d'années) sont célèbres mondialement pour au moins trois raisons principales. D'abord parce que les continents sont alors regroupés en une seule masse émergée, la Pangée. Cette dernière est soumise à un climat caractérisé par une forte tendance à l'aridité.

Ensuite parce que la limite Permien-Trias a été définie par la plus grande extinction biologique de tous les temps et durant laquelle près de 90% des espèces ont disparu. Cette extinction majeure fait l'objet de nombreuses recherches et permet de mettre en perspective l'érosion actuelle de la biodiversité. Enfin, parce que le Trias est un intervalle durant lequel le monde vivant se diversifie à nouveau et voit l'apparition de nombreux groupes de vertébrés. Deux d'entre eux sont emblématiques et connus de tous: les dinosaures et les mammifères.

Ils sont accompagnés de nombreux autres clades qui auront soit une durée de vie géologique fugace (aétosaures, phytosaures, rhynchosaures), soit au contraire une étendue temporelle plus longue (tortues, crocodiles, squamates, lissamphibiens). Ce profond renouvellement faunique est lui-même en partie associé à une crise d'extinction majeure affectant particulièrement les faunes terrestres au Trias Supérieur (passage Carnien-Norien, il y a 227 millions d'années) mais qui permettra aux dinosaures d'asseoir leur domination dans les écosystèmes terrestres.

Dans ce contexte, le département de l'Ardèche est remarquable car il offre un accès privilégié aux roches sédimentaires permo-triasiques dans lesquelles abondent les traces de vertébrés fossiles [START_REF] Montenat | Empreintes de pas de reptiles dans le Trias moyen du plateau du Daüs près d'Aubenas (Ardèche)[END_REF][START_REF] Courel | Une ichnofaune reptilienne remarquable dans les Grès triasiques de Largentière Ardèche, France[END_REF][START_REF] Courel | Une nouvelle ichnoespèce Coelurosaurichnus grancieri du Trias supérieur de l'Ardèche, France[END_REF][START_REF] Gand | Les empreintes animales et végétales du bassin permien de Largentière Ardèche, France. Conséquences stratigraphiques et paléoenvironnementales[END_REF][START_REF] Gand | Les traces dinosauroïdes du Trias supérieur français : discrimination, interprétation et comparaison[END_REF][START_REF] Gand | Les traces de pas d'Amphibiens, de Dinosaures et autres Reptiles du Mésozoique français : Inventaire et interprétations[END_REF][START_REF] Gand | Les pistes dinosauroïdes du Trias moyen français: interprétation et réévaluation de la nomenclature[END_REF]. Plus largement, en préservant une diversité lithologique composée de roches plutoniques, métamorphiques, volcaniques et sédimentaires, l'Ardèche rassemble des structures géologiques remarquables et des témoignages de leur utilisation par l'homme (exploitations minières, constructions…). Cette situation exceptionnelle a notamment permis au Parc Naturel Régional (PNR) des Monts d'Ardèche d'obtenir le label Géoparc mondial de l'UNESCO en 2014. Ce label implique notamment de renforcer l'articulation avec la recherche scientifique et de promouvoir la diffusion des savoirs. Dans le domaine de la palichnologie (étude des traces fossiles), cette mission a été confiée à l'Université de Bourgogne (Laboratoire Biogéosciences, Dijon). En effet, depuis la fin des années 1960, cette institution académique a largement contribué à la description scientifique du patrimoine ichnologique ardéchois [START_REF] Gand | Les pistes dinosauroïdes du Trias moyen français: interprétation et réévaluation de la nomenclature[END_REF][START_REF] Gand | Les traces de pas d'Amphibiens, de Dinosaures et autres Reptiles du Mésozoique français : Inventaire et interprétations[END_REF][START_REF] Gand | Les pistes dinosauroïdes du Trias moyen français: interprétation et réévaluation de la nomenclature[END_REF][START_REF] Szewczyk | Tracking dinosaurs in coarse-grained sediments from the Upper Triassic of Ardèche (southeastern France)[END_REF][START_REF]Inventory and conservation of Triassic vertebrate tracks in the Monts d'Ardèche UNESCO Global Geopark, France[END_REF]

Contexte géologique

Les roches sédimentaires permiennes et triasiques ardéchoises affleurent principalement le long d'une bande NE-SW entre Privas et Les Vans. Dans la série stratigraphique de la bordure orientale du Massif Central, quatre unités majeures livrent l'essentiel des traces (ichnites) de vertébrés (Figure 1). Figure 1 Pour le Permien, les ichnites proviennent de la Formation de la Lande décrite par [START_REF] Gand | Les empreintes animales et végétales du bassin permien de Largentière Ardèche, France. Conséquences stratigraphiques et paléoenvironnementales[END_REF], correspondant à l'unité IV de la Séquence III de Becq-Giraudon (1993). Cette unité est datée du Permien basal (« Autunien ») et correspond à des paléoenvironnements continentaux [START_REF] Gand | Les empreintes animales et végétales du bassin permien de Largentière Ardèche, France. Conséquences stratigraphiques et paléoenvironnementales[END_REF].

Pour le Trias, les trois unités majeures sont la Formation des Grès du Roubreau datée du Trias Moyen (Anisien/Ladinien), et le duo formé par l'Ensemble gréso-dolomitique gris et la Formation Bariolée d'Ucel (sensu [START_REF] Gand | Les traces de pas d'Amphibiens, de Dinosaures et autres Reptiles du Mésozoique français : Inventaire et interprétations[END_REF] [START_REF] Montenat | Empreintes de pas de reptiles dans le Trias moyen du plateau du Daüs près d'Aubenas (Ardèche)[END_REF][START_REF] Courel | Une ichnofaune reptilienne remarquable dans les Grès triasiques de Largentière Ardèche, France[END_REF][START_REF] Courel | Une nouvelle ichnoespèce Coelurosaurichnus grancieri du Trias supérieur de l'Ardèche, France[END_REF]. Bien que la validité de certaines de ces références ichnologiques internationales doive être réévaluée [START_REF] Gand | Les pistes dinosauroïdes du Trias moyen français: interprétation et réévaluation de la nomenclature[END_REF], elles démontrent elles aussi le remarquable potentiel ichnologique de la série permo-triasique ardéchoise.

Richesse ichnologique et auteurs présumés

À ce jour, plus de 20 localités et près de 1500 empreintes individuelles ont été inventoriées le long de la succession permo-triasique ardéchoise (figure 1). Il ne fait aucun doute qu'il reste encore d'innombrables ichnites à étudier et à découvrir en Ardèche, comme en témoigne la mise au jour de nouvelles empreintes lors de chaque mission de prospection.

Nous dressons ici un bref état des lieux des types d'empreintes de vertébrés permotriasiques identifiés dans ce département, ainsi qu'un panorama des auteurs présumés de ces ichnites. À cette occasion, il est utile de rappeler qu'en ichnologie, il est rarement possible de déterminer l'auteur précis d'une empreinte. En effet, des espèces différentes peuvent laisser des empreintes similaires, et une même espèce peut laisser divers types d'empreintes. L'identification de l'auteur revient donc à mener une enquête visant à déterminer une sorte de portrait-robot, ce qui consiste en pratique à déduire un ou plusieurs grands groupes d'organismes. Afin de faciliter la lecture et de mettre en avant des groupes de reptiles parfois méconnus, nous présentons ces derniers sur un arbre de parenté simplifié (Figure 2). Un tel mode de préservation est rarissime car il s'écarte des modèles de préservation dominants qui impliquent des processus microbiens [START_REF] Szewczyk | Tracking dinosaurs in coarse-grained sediments from the Upper Triassic of Ardèche (southeastern France)[END_REF]. Il illustre également le potentiel de préservation contrasté des « vraies traces » (empreintes laissées sur la surface de marche réelle), des « sous-traces » (empreintes résultant de la déformation des couches sédimentaires sous la surface de marche) et des « sur-traces » (empreintes correspondant à une couverture sédimentaire peu profonde des vraies traces après leur formation). [START_REF] Laurin | A reevaluation of early amniote phylogeny[END_REF], [START_REF] Nesbitt | The early evolution of Archosauria: relationships and the origin of major clades[END_REF][START_REF] Ezcurra | The phylogenetic relationships of basal archosauromorphs, with an emphasis on the systematics of proterosuchian archosauriforms[END_REF]. Les silhouettes ne sont pas représentées à l'échelle.

Conservation et valorisation du patrimoine ichnologique ardéchois

  . Ce partenariat de longue date a été formellement concrétisé en novembre 2015 par une convention entre le PNR des Monts d'Ardèche et l'Université de Bourgogne. La convention s'articule autour de quatre volets : (1) la recherche scientifique, (2) la protection des sites ichnologiques, (3) la formation des étudiants et des professionnels et (4) la vulgarisation scientifique. Les principaux objectifs sont de réaliser un inventaire détaillé et sédimentologiquement contextualisé des traces de vertébrés du Permien et du Trias au sein du Géoparc et de ses environs, d'établir des priorités de conservation et de protection, et de contribuer à mettre en place des géosites ichnologiques pour le public (Fara et al. 2021). Cette contribution a comme ambition de présenter brièvement le contexte, les enjeux et les perspectives de l'étude des traces fossiles de vertébrés du Permien et du Trias d'Ardèche.

  datée du Trias Supérieur (Carnien supérieur/Norien). La Formation des Grès du Roubreau est dominée par des conglomérats et des grès, tandis que des grès, des microconglomérats et des couches d'argilite occasionnelles forment l'essentiel de l'Ensemble gréso-dolomitique gris et de la Formation Bariolée d'Ucel. Ces unités continentales illustrent des environnements fluviatiles et de plaines d'inondation. Elles ont été datées palynologiquement. Le patrimoine ichnologique livré par ces unités triasiques est connu mondialement et il est largement dominé par les empreintes de vertébrés, notamment les reptiles. Certaines d'entre elles ont fait l'objet d'une description scientifique et ont permis l'érection d'espèces nouvelles d'empreintes également appelées ichnoespèces, telles que Isochirotherium felenci, Isochirotherium delicatum, Coelurosaurichnus grancieri, Coelurosaurichnus largentierensis et Dahutherium agilis qui ont été laissées par des reptiles affiliés à la lignée des crocodiles et des dinosaures

Figure 2 Formation

 2 Figure 2

Figure 3 Les

 3 Figure3

Figure 4 L

 4 Figure 4

Figure 5

 5 Figure5

  Aménagement de sites Le géosite du Sartre est le premier géosite ichnologique d'Ardèche. Découvert par l'un de nous (MV), et après quatre années d'investigations scientifiques et de travaux d'aménagement, ce site a été inauguré en octobre 2020 (Figure 6). Il s'agit du premier site public aménagé en France (et probablement en Europe) présentant des traces de dinosaures du Trias supérieur. Les visiteurs ont librement accès à des panneaux explicatifs illustrés, à des tubes de ciblage visuel et à des répliques 3D d'empreintes reptiliennes. Sous la coordination du PNR/Géoparc des Monts d'Ardèche, ce premier succès dans l'aménagement d'un ichnosite est le fruit d'une synergie remarquable entre une multitude d'acteurs locaux, d'associations, de collectivités territoriales, d'institutions académiques et de programmes de financements français et européens (Fara et al. 2021). Le site permien des Fourniols à Largentière et un parcours balisé dans le Trias de Tauriers s'inscrivent également dans un projet de valorisation. Ces aménagements, plus ou moins conséquents, ouvrent des perspectives enthousiasmantes de « circuits découvertes » ou « balades géologiques » en terres ardéchoises, notamment dans les secteurs de Sanilhac, de Tauriers et de Largentière.

Figure 6 Figure 1 :

 61 Figure 6

Figure 2 :

 2 Figure2: Arbre de parenté simplifié des vertébrés amniotes montrant les groupes auteurs des traces permo-triasiques d'Ardèche. Les relations de parenté sont basées sur les travauxde Laurin & Reisz (1995),[START_REF] Nesbitt | The early evolution of Archosauria: relationships and the origin of major clades[END_REF][START_REF] Ezcurra | The phylogenetic relationships of basal archosauromorphs, with an emphasis on the systematics of proterosuchian archosauriforms[END_REF]. Les silhouettes ne sont pas représentées à l'échelle.

Figure 3 :

 3 Figure 3 : Traces de vertébrés permiens de la Formation de la Lande, au nord-est de Largentière. L'encart montre une empreinte de pied droit nommée Dimetropus leisnerianus et laissée par un pélycosaure (« reptile mammalien »). Tracé d'empreinte d'après Gand(1994).

Figure 4 :

 4 Figure 4 : Empreintes de pas de reptiles pseudosuchiens laissées dans la Formation des Grès du Roubreau (Trias moyen) : Isochirotherium felenci (A et B) et Isochirotherium

Figure 5 :

 5 Figure 5 : Dalle à Coelurosaurichnus grancieri trouvée par Max Grancier dans l'Ensemble gréso-dolomitique gris (Trias supérieur) près de Payzac (A) et empreinte de type Grallator isp. dans la Formation Bariolée d'Ucel au géosite du Sartre (commune d'Ucel) illustrant la granulométrie grossière des roches sédimentaires (B).

Figure 6 :

 6 Figure 6 : Le géosite ichnologique du Sartre (commune d'Ucel) juste avant son inauguration en octobre 2020. Il s'agit du premier site public entièrement aménagé en Europe présentant des traces de dinosaures du Trias supérieur.

  FIGURE 1

Remerciements

Nous tenons à remercier la Société Géologique de l'Ardèche (notamment Nicolas Gignoux et Dominique Gasquet pour leur relecture du manuscrit), tous les habitants et bénévoles qui nous aident à promouvoir le patrimoine ichnologique ardéchois, en particulier Maryse