**Lemma 10.3.** For $x \in \mathbb{T}^r$ and $y_1^{\pm 1}, \dots, y_n^{\pm 1}$ monomials as in Lemma 10.2, let $C \in \mathbb{R}^{n \times n}$ be a matrix with eigenvalues $(y_i(x) + y_i(x)^{-1})/2$. Then, for $k \in \mathbb{N}$, the trace of the $k$-th power of $C$ is

$$ \mathrm{Tr}(C^k) = \frac{n}{2^k} \sum_{\ell=0}^{[k/2]-1} \left( \binom{k}{\ell} (\Theta_{(k-2\ell)e_1}(x) + \Theta_{(2\ell-k)e_1}(x)) \right) + \frac{n}{2^k} \begin{cases} \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases}. $$

*Proof.* The trace of the $k$-th power of $C$ is the sum of the $k$-th powers of the eigenvalues. Thus,

$$ \begin{aligned} \mathrm{Tr}(C^k) &= \sum_{i=1}^{n} \left( \frac{y_i(x) + y_i(x)^{-1}}{2} \right)^k = \frac{1}{2^k} \sum_{i=1}^{n} \left( \sum_{\ell=0}^{k} \binom{k}{\ell} y_i(x)^{k-\ell} y_i(x)^{-\ell} \right) \\ &= \frac{1}{2^k} \sum_{i=1}^{n} \left( \sum_{\ell=0}^{[k/2]-1} \binom{k}{\ell} y_i(x)^{k-2\ell} + \sum_{\ell=[k/2]+1}^{k} \binom{k}{\ell} y_i(x)^{k-2\ell} + \begin{cases} \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases} \right) \\ &= \frac{1}{2^k} \sum_{\ell=0}^{[k/2]-1} \left( \binom{k}{\ell} \sum_{i=1}^{n} (y_i(x)^{k-2\ell} + y_i(x)^{2\ell-k}) \right) + \frac{n}{2^k} \begin{cases} \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases}. \end{aligned} $$

Applying Lemma 10.2 yields the formula. $\square$

*Proof of Theorem 10.1.* The $\mathbb{T}$-orbit space of $\mathcal{W}$ is by definition $\mathcal{T} = \{\vartheta(x) | x \in \mathbb{T}^r\}$. According to Theorems 5.3, 6.3, 7.4, 8.4 and 9.3, we have $\mathcal{T} = \{z \in \mathcal{Z} | H(z) \ge 0\}$. Here, $\mathcal{Z}$ is an $\mathbb{R}$-vector space with $\dim(\mathcal{Z}) = r$ and $H \in \mathbb{Q}[z]^{n \times n}$ is a matrix with entries $H_{ij} = \mathrm{Tr}(C^{i+j-2}) - \mathrm{Tr}(C^{i+j})$, where $C \in \mathbb{Q}[z]^{n \times n}$ is defined in the respective theorem.

For $x \in \mathbb{T}^r$ and $z := \vartheta(x) \in \mathcal{T} \subseteq \mathcal{Z}$, it is now essential to note that $C(z) \in \mathbb{R}^{n \times n}$ has eigenvalues $(y_i(x) + y_i(x)^{-1})/2 \in [-1, 1]$ with $y_1^{\pm 1}, \dots, y_n^{\pm 1}$ 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

$$ \Theta_{(k-2\ell)e_1}(x) + \Theta_{(2\ell-k)e_1}(x) = T_{(k-2\ell)e_1}(\vartheta(x)) + T_{(2\ell-k)e_1}(\vartheta(x)) = 2 \hat{T}_{(k-2\ell)e_1}(\hat{\vartheta}(x)). $$

We now substitute $z \mapsto \hat{z} := \hat{\vartheta}(x) \in \hat{\mathcal{T}} \subseteq \mathbb{R}^n$. For $1 \le i, j \le n$ and $k := i + j$, the entry $H(z)_{ij}$ is

$$ \begin{aligned} & \mathrm{Tr}(C(z)^{k-2}) - \mathrm{Tr}(C(z)^k) \\ &= \frac{2n}{2^k} \left( 4 \sum_{\ell=0}^{[k/2]-2} \left( \binom{k-2}{\ell} \hat{T}_{(k-2(\ell+1))e_1}(\hat{z}) \right) - \sum_{\ell=0}^{[k/2]-1} \left( \binom{k}{\ell} \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) \right) \\ & \quad + \frac{n}{2^k} \begin{cases} 4 \binom{k-2}{k/2-1} - \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases} \\ &= \frac{2n}{2^k} \left( 4 \sum_{\ell=1}^{[k/2]-1} \left( \binom{k-2}{\ell-1} \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) - \sum_{\ell=0}^{[k/2]-1} \left( \binom{k}{\ell} \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) \right) \\ & \quad + \frac{n}{2^k} \begin{cases} 4 \binom{k-2}{k/2-1} - \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases} \\ &= \frac{2n}{2^k} \left( -\hat{T}_{k e_1}(\hat{z}) + \sum_{\ell=1}^{[k/2]-1} \left( 4 \binom{k-2}{\ell-1} - \binom{k}{\ell} \right) \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) + \frac{n}{2^k} \begin{cases} 4 \binom{k-2}{k/2-1} - \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases}. \end{aligned} $$

Dividing by $2n$ does not change whether $H(\hat{z})$ is positive semi-definite and so we obtain the formula. $\square$
**Lemma 10.3.** For $x \in \mathbb{T}^r$ and $y_1^{\pm 1}, \dots, y_n^{\pm 1}$ monomials as in Lemma 10.2, let $C \in \mathbb{R}^{n \times n}$ be a matrix with eigenvalues $(y_i(x) + y_i(x)^{-1})/2$. Then, for $k \in \mathbb{N}$, the trace of the $k$-th power of $C$ is

$$ \mathrm{Tr}(C^k) = \frac{n}{2^k} \sum_{\ell=0}^{\lfloor k/2 \rfloor - 1} \left( \binom{k}{\ell} (\Theta_{(k-2\ell)e_1}(x) + \Theta_{(2\ell-k)e_1}(x)) \right) + \frac{n}{2^k} \begin{cases} \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases}. $$

*Proof.* The trace of the $k$-th power of $C$ is the sum of the $k$-th powers of the eigenvalues. Thus,

$$ \begin{aligned} \mathrm{Tr}(C^k) &= \sum_{i=1}^{n} \left( \frac{y_i(x) + y_i(x)^{-1}}{2} \right)^k = \frac{1}{2^k} \sum_{i=1}^{n} \left( \sum_{\ell=0}^{k} \binom{k}{\ell} y_i(x)^{k-\ell} y_i(x)^{-\ell} \right) \\ &= \frac{1}{2^k} \sum_{i=1}^{n} \left( \sum_{\ell=0}^{\lfloor k/2 \rfloor - 1} \binom{k}{\ell} y_i(x)^{k-2\ell} + \sum_{\ell=\lfloor k/2 \rfloor + 1}^{k} \binom{k}{\ell} y_i(x)^{k-2\ell} + \begin{cases} \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases} \right) \\ &= \frac{1}{2^k} \sum_{\ell=0}^{\lfloor k/2 \rfloor - 1} \left( \binom{k}{\ell} \sum_{i=1}^{n} (y_i(x)^{k-2\ell} + y_i(x)^{2\ell-k}) \right) + \frac{n}{2^k} \begin{cases} \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases}. \end{aligned} $$

Applying Lemma 10.2 yields the formula. $\square$

*Proof of Theorem 10.1.* The $\mathbb{T}$-orbit space of $\mathcal{W}$ is by definition $\mathcal{T} = \{\vartheta(x) | x \in \mathbb{T}^r\}$. According to Theorems 5.3, 6.3, 7.4, 8.4 and 9.3, we have $\mathcal{T} = \{z \in \mathcal{Z} | H(z) \succeq 0\}$. Here, $\mathcal{Z}$ is an $\mathbb{R}$-vector space with $\dim(\mathcal{Z}) = r$ and $H \in \mathbb{Q}[z]^{n \times n}$ is a matrix with entries $H_{ij} = \mathrm{Tr}(C^{i+j-2}) - \mathrm{Tr}(C^{i+j})$, where $C \in \mathbb{Q}[z]^{n \times n}$ is defined in the respective theorem.

For $x \in \mathbb{T}^r$ and $z := \vartheta(x) \in \mathcal{T} \subseteq \mathcal{Z}$, it is now essential to note that $C(z) \in \mathbb{R}^{n \times n}$ has eigenvalues $(y_i(x) + y_i(x)^{-1})/2 \in [-1, 1]$ with $y_1^{\pm 1}, \dots, y_n^{\pm 1}$ 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

$$ \Theta_{(k-2\ell)e_1}(x) + \Theta_{(2\ell-k)e_1}(x) = T_{(k-2\ell)e_1}(\vartheta(x)) + T_{(2\ell-k)e_1}(\vartheta(x)) = 2 \hat{T}_{(k-2\ell)e_1}(\hat{\vartheta}(x)). $$

We now substitute $z \mapsto \hat{z} := \hat{\vartheta}(x) \in \hat{\mathcal{T}} \subseteq \mathbb{R}^n$. For $1 \le i, j \le n$ and $k := i + j$, the entry $H(z)_{ij}$ is

$$ \begin{aligned} & \mathrm{Tr}(C(z)^{k-2}) - \mathrm{Tr}(C(z)^k) \\ &= \frac{2n}{2^k} \left( 4 \sum_{\ell=0}^{\lfloor k/2 \rfloor - 2} \left( \binom{k-2}{\ell} \hat{T}_{(k-2(\ell+1))e_1}(\hat{z}) \right) - \sum_{\ell=0}^{\lfloor k/2 \rfloor - 1} \left( \binom{k}{\ell} \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) \right) \\ & \quad + \frac{n}{2^k} \begin{cases} 4 \binom{k-2}{k/2-1} - \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases} \\ &= \frac{2n}{2^k} \left( 4 \sum_{\ell=1}^{\lfloor k/2 \rfloor - 1} \left( \binom{k-2}{\ell-1} \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) - \sum_{\ell=0}^{\lfloor k/2 \rfloor - 1} \left( \binom{k}{\ell} \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) \right) \\ & \quad + \frac{n}{2^k} \begin{cases} 4 \binom{k-2}{k/2-1} - \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases} \\ &= \frac{2n}{2^k} \left( -\hat{T}_{k e_1}(\hat{z}) + \sum_{\ell=1}^{\lfloor k/2 \rfloor - 1} \left( 4 \binom{k-2}{\ell-1} - \binom{k}{\ell} \right) \hat{T}_{(k-2\ell)e_1}(\hat{z}) \right) + \frac{n}{2^k} \begin{cases} 4 \binom{k-2}{k/2-1} - \binom{k}{k/2}, & k \text{ even} \\ 0, & k \text{ odd} \end{cases}. \end{aligned} $$

Dividing by $2n$ does not change whether $H(\hat{z})$ is positive semi-definite and so we obtain the formula. $\square$
**Remark 10.4.** The matrix polynomial $H \in \mathbb{Q}[\hat{z}]^{n \times n}$ from Theorem 10.1 follows the pattern

$$ \begin{bmatrix} \frac{\hat{T}_0 - \hat{T}_{2e_1}}{4} & \frac{\hat{T}_{e_1} - \hat{T}_{3e_1}}{8} & \frac{\hat{T}_{0} - \hat{T}_{4e_1}}{16} & \frac{2\hat{T}_{e_1} - \hat{T}_{3e_1} - \hat{T}_{5e_1}}{32} & \dots \\ \frac{\hat{T}_{e_1} - \hat{T}_{3e_1}}{8} & \frac{\hat{T}_{0} - \hat{T}_{4e_1}}{16} & \frac{2\hat{T}_{e_1} - \hat{T}_{3e_1} - \hat{T}_{5e_1}}{32} & \frac{2\hat{T}_{0} + \hat{T}_{2e_1} - 2\hat{T}_{4e_1} - \hat{T}_{6e_1}}{64} & \dots \\ \frac{2\hat{T}_{e_1} - \hat{T}_{3e_1} - \hat{T}_{5e_1}}{32} & \frac{2\hat{T}_{0} + \hat{T}_{2e_1} - 2\hat{T}_{4e_1} - \hat{T}_{6e_1}}{64} & \frac{5\hat{T}_{e_1} - \hat{T}_{3e_1} - 3\hat{T}_{5e_1} - \hat{T}_{7e_1}}{128} & \frac{5\hat{T}_{0} + 4\hat{T}_{2e_1} - 4\hat{T}_{4e_1} - 4\hat{T}_{6e_1} - \hat{T}_{8e_1}}{256} & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} $$

1. The only $\hat{T}_{\alpha}$ occurring in $H$ are those, where $\alpha \in \mathbb{N}^r$ 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 $2n = \deg(\hat{T}_{2n e_1})$.

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 $2n-1$.

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 $k \le n$.

## Conclusion and future work

We have characterized the $\mathcal{T}$-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.

## 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 Skłodowska-Curie Actions, grant agreement 813211 (POEMA). Minor changes were applied during his postdoctoral research at RPTU Kaiserslautern-Landau, supported by the Deutsche Forschungs-gemeinschaft transregional collaborative research centre (SFB-TRR) 195 "Symbolic Tools in Mathematics and their Application".

## References

[BDOV14] C. Bachoc, E. Decorte, F. de Oliviera Filho, and F. Vallentin. Spectral bounds for the in-dependence ratio and the chromatic number of an operator. *Israel Journal of Mathematics*, 202(1):227-254, 2014.

[Bee91] R. Beerends. Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator. *Transactions of the American Mathematical Society*, 328(2):779-814, 1991.
**Remark 10.4.** The matrix polynomial $H \in \mathbb{Q}[\hat{z}]^{n \times n}$ from Theorem 10.1 follows the pattern

$$ \begin{bmatrix} \frac{\hat{T}_0 - \hat{T}_{2e_1}}{4} & \frac{\hat{T}_{e_1} - \hat{T}_{3e_1}}{8} & \frac{\hat{T}_{0} - \hat{T}_{4e_1}}{16} & \frac{2\hat{T}_{e_1} - \hat{T}_{3e_1} - \hat{T}_{5e_1}}{32} & \dots \\ \frac{\hat{T}_{e_1} - \hat{T}_{3e_1}}{8} & \frac{\hat{T}_{0} - \hat{T}_{4e_1}}{16} & \frac{2\hat{T}_{e_1} - \hat{T}_{3e_1} - \hat{T}_{5e_1}}{32} & \frac{2\hat{T}_{0} + \hat{T}_{2e_1} - 2\hat{T}_{4e_1} - \hat{T}_{6e_1}}{64} & \dots \\ \frac{2\hat{T}_{e_1} - \hat{T}_{3e_1} - \hat{T}_{5e_1}}{32} & \frac{2\hat{T}_{0} + \hat{T}_{2e_1} - 2\hat{T}_{4e_1} - \hat{T}_{6e_1}}{64} & \frac{5\hat{T}_{e_1} - \hat{T}_{3e_1} - 3\hat{T}_{5e_1} - \hat{T}_{7e_1}}{128} & \frac{5\hat{T}_{0} + 4\hat{T}_{2e_1} - 4\hat{T}_{4e_1} - 4\hat{T}_{6e_1} - \hat{T}_{8e_1}}{256} & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} $$

1. The only $\hat{T}_{\alpha}$ occurring in $H$ are those, where $\alpha \in \mathbb{N}^r$ 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 $2n = \text{deg}(\hat{T}_{2n} e_1)$.

2. $H$ has Hankel structure, that is, $H_{ij} = H_{k\ell}$ whenever $i+j=k+\ell$. Therefore, the number of distinct entries is at most $2n-1$.

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 $k \le n$.

## Conclusion and future work

We have characterized the $\mathcal{T}$-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.

## 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 Skłodowska-Curie Actions, grant agreement 813211 (POEMA). Minor changes were applied during his postdoctoral research at RPTU Kaiserslautern-Landau, supported by the Deutsche Forschungs-gemeinschaft transregional collaborative research centre (SFB-TRR) 195 "Symbolic Tools in Mathematics and their Application".

## References

[BDOV14] C. Bachoc, E. Decorte, F. de Oliviera Filho, and F. Vallentin. Spectral bounds for the in-dependence ratio and the chromatic number of an operator. *Israel Journal of Mathematics*, 202(1):227-254, 2014.

[Bee91] R. Beerends. Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator. *Transactions of the American Mathematical Society*, 328(2):779-814, 1991.
[Bou68] N. Bourbaki. *Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Titres. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines.* Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris, 1968.

[BPT13] G. Blekherman, P. Parrilo, and R. Thomas, editors. *Semidefinite optimization and convex algebraic geometry*, volume 13 of *MOS/SIAM Ser. Optim*. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2013.

[BWFW08] X. Bai, H. Wei, K. Fujisawa, and Y. Wang. *Semidefinite programming for optimal power flow problems*. *International Journal of Electrical Power & Energy Systems*, 30(6):383-392, 2008.

[CLO05] D. Cox, J. Little, and D. O'Shea. *Using algebraic geometry*, volume 185 of *Graduate Texts in Mathematics*. Springer, New York, second edition, 2005.

[CLO15] D. Cox, J. Little, and D. O'Shea. *Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra*. Undergraduate Texts in Mathematics. Springer, New York, NY, 2015.

[CS99] J. Conway and N. Sloane. *Sphere packings, lattices and groups*, volume 290 of *Grundlehren der Mathematischen Wissenschaften*. Springer-Verlag, New York, third edition, 1999.

[DL80] K. Dunn and R. Lidl. *Multi-dimensional generalizations of the Chebyshev polynomials*, I, II. *Proceedings of the Japan Academy*, 56:154-165, 1980.

[DM85] H. Dym and H. McKean. *Fourier Series and Integrals*. Probability and Mathematical Statistics. Elsevier Science, 1985.

[Dum07] B. Dumitrescu. *Positive Trigonometric Polynomials and Signal Processing Applications*. Signals and Communication Technology. Springer Netherlands, 2007.

[EL82] R. Eier and R. Lidl. *A class of orthogonal polynomials in k variables*. *Mathematische Annalen*, 260:93-100, 1982.

[Far84] D. Farkas. *Multiplicative invariants*. *L'Enseignement Mathématique*, 30(1-2):141-157, 1984.

[HHM16] L. Háková, J. Hrivnák, and L. Motlochová. *On cubature rules associated to Weyl group orbit functions*. *Acta Polytechnica*, 56(3):202-213, 2016.

[HMMR22] E. Hubert, T. Metzlaff, P. Moustrou, and C. Riener. *T-orbit spaces of multiplicative actions and applications*. *ACM Communications in Computer Algebra*, 56(2):72-75, 2022.

[HMMR23] E. Hubert, T. Metzlaff, P. Moustrou, and C. Riener. *Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs*. Preprint hal-03768067, 2023.

[HS22] E. Hubert and M. Singer. *Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials*. *Foundations of Computational Mathematics*, 22:1801-1862, 2022.

[HW88] M. Hoffman and W. Withers. *Generalized Chebyshev polynomials associated with affine Weyl groups*. *Transactions of the American Mathematical Society*, 308(1):91-104, 1988.

[JM18] C. Josz and D. Molzahn. *Lasserre hierarchy for large scale polynomial optimization in real and complex variables*. *SIAM Journal of Optimization*, 28(2):1017-1048, 2018.

[Koo74] T. Koornwinder. *Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators*. *Indagationes Mathematicae (Proceedings)*, 77(1):48-58, 1974.
[Bou68] N. Bourbaki. *Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Titres. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines.* Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris, 1968.

[BPT13] G. Blekherman, P. Parrilo, and R. Thomas, editors. *Semidefinite optimization and convex algebraic geometry*, volume 13 of *MOS/SIAM Ser. Optim*. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2013.

[BWFW08] X. Bai, H. Wei, K. Fujisawa, and Y. Wang. Semidefinite programming for optimal power flow problems. *International Journal of Electrical Power & Energy Systems*, 30(6):383-392, 2008.

[CLO05] D. Cox, J. Little, and D. O'Shea. *Using algebraic geometry*, volume 185 of *Graduate Texts in Mathematics*. Springer, New York, second edition, 2005.

[CLO15] D. Cox, J. Little, and D. O'Shea. *Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra*. Undergraduate Texts in Mathematics. Springer, New York, NY, 2015.

[CS99] J. Conway and N. Sloane. *Sphere packings, lattices and groups*, volume 290 of *Grundlehren der Mathematischen Wissenschaften*. Springer-Verlag, New York, third edition, 1999.

[DL80] K. Dunn and R. Lidl. Multi-dimensional generalizations of the Chebyshev polynomials, I, II. *Proceedings of the Japan Academy*, 56:154-165, 1980.

[DM85] H. Dym and H. McKean. *Fourier Series and Integrals*. Probability and Mathematical Statistics. Elsevier Science, 1985.

[Dum07] B. Dumitrescu. *Positive Trigonometric Polynomials and Signal Processing Applications*. Signals and Communication Technology. Springer Netherlands, 2007.

[EL82] R. Eier and R. Lidl. A class of orthogonal polynomials in $k$ variables. *Mathematische Annalen*, 260:93-100, 1982.

[Far84] D. Farkas. Multiplicative invariants. *L'Enseignement Mathématique*, 30(1-2):141-157, 1984.

[HHM16] L. Háková, J. Hrivnák, and L. Motlochová. On cubature rules associated to Weyl group orbit functions. *Acta Polytechnica*, 56(3):202-213, 2016.

[HMMR22] E. Hubert, T. Metzlaff, P. Moustrou, and C. Riener. T-orbit spaces of multiplicative actions and applications. *ACM Communications in Computer Algebra*, 56(2):72-75, 2022.

[HMMR23] E. Hubert, T. Metzlaff, P. Moustrou, and C. Riener. Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs. Preprint hal-03768067, 2023.

[HS22] E. Hubert and M. Singer. Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials. *Foundations of Computational Mathematics*, 22:1801-1862, 2022.

[HW88] M. Hoffman and W. Withers. Generalized Chebyshev polynomials associated with affine Weyl groups. *Transactions of the American Mathematical Society*, 308(1):91-104, 1988.

[JM18] C. Josz and D. Molzahn. Lasserre hierarchy for large scale polynomial optimization in real and complex variables. *SIAM Journal of Optimization*, 28(2):1017-1048, 2018.

[Koo74] T. Koornwinder. Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. *Indagationes Mathematicae (Proceedings)*, 77(1):48-58, 1974.
[Las01] J.-B. Lasserre. Global Optimization with Polynomials and the Problem of Moments. *SIAM Journal of Optimization*, 11(3):796-817, 2001.

[Lau09] M. Laurent. *Sums of Squares, Moment Matrices and Optimization Over Polynomials*, pages 157-270. Springer New York, New York, NY, 2009.

[Lor05] M. Lorenz. *Multiplicative Invariant Theory*. Encyclopaedia of Mathematical Sciences. Springer, Berlin, Heidelberg, 2005.

[LSX08] H. Li, J. Sun, and Y. Xu. Discrete Fourier analysis, cubature, and interpolation on a hexagon and triangle. *SIAM Journal of Numerical Analysis*, 46(4):1653-1681, 2008.

[LSX12] H. Li, J. Sun, and Y. Xu. Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group. *SIGMA. Symmetry, Integrability and Geometry. Methods and Applications*, 8:67-96, 2012.

[LX10] H. Li and Y. Xu. Discrete Fourier analysis on fundamental domain and simplex of $A_d$ lattice in $d$ variables. *The Journal of Fourier Analysis and Applications*, 16(3):383-433, 2010.

[Mac90] I. MacDonald. Orthogonal polynomials associated with root systems. In *Orthogonal Polynomials: Theory and Practice*, volume 294 of *NATO ASI Series*, pages 311-318. Springer, Netherlands, Dordrecht, 1990.

[Met22] T. Metzlaff. *Groupes Cristallographiques et Polynômes de Chebyshev en Optimisation Globale*. Thèse de doctorat en Mathématiques, dirigée par E. Hubert, Inria d'Université Côte d'Azur, theses.fr/2022COAZ4094, 2022.

[MNR13] H. Munthe-Kaas, M. Nome, and B. Ryland. Through the kaleidoscope: symmetries, groups and Chebyshev-approximations from a computational point of view. In *Foundations of computational mathematics*, Budapest 2011, London Mathematical Society Lecture Note Series, pages 188-229. Cambridge Univ. Press, Cambridge, 2013.

[MP11] R. Moody and J. Patera. Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups. *Advances in Applied Mathematics*, 47:509-535, 2011.

[Pro78] C. Procesi. Positive symmetric functions. *Advances in Mathematics*, 29(2):219-225, 1978.

[PS85] C. Procesi and G. Schwarz. Inequalities defining orbit spaces. *Inventiones mathematicae*, 81:539-554, 1985.

[Ste75] R. Steinberg. On a theorem of Pittie. *Topology*, 14(2):173-177, 1975.

[Tal20] V. Talamini. Generating formulas for finite reflection groups of the infinite series $Sn$, $An$, $Bn$ and $Dn$. *Rendiconti del Circolo Matematico di Palermo Series 2*, 69:1043-1077, 2020.

[Xu15] Y. Xu. Generalized characteristic polynomials and Gaussian cubature rules. *SIAM Journal on Matrix Analysis and Applications*, 36:1129-1142, 2015.

E. Hubert, CENTRE INRIA D'UNIVERSITÉ CÔTE D'AZUR, 06902 SOPHIA ANTIPOLIS, FRANCE

E-mail address: evelyne.hubert@inria.fr

ORCID: 0000-0003-1456-9524

T. Metzlaff, DEPARTMENT OF MATHEMATICS, RPTU KAISERSLAUTERN-LANDAU, 67663 KAISERSLAUTERN, GERMANY

Former Affiliation: CENTRE INRIA D'UNIVERSITÉ CÔTE D'AZUR, 06902 SOPHIA ANTIPOLIS, FRANCE

E-mail address: tobias.metzlaff@rptu.de

ORCID: 0000-0002-0688-7074

C. Riener, DEPARTMENT OF MATHEMATICS, UIT THE ARCTIC UNIVERSITY, 9037 TROMSØ, NORWAY

E-mail address: cordian.riener@uit.no

ORCID: 0000-0002-1192-3500
[Las01] J.-B. Lasserre. Global Optimization with Polynomials and the Problem of Moments. *SIAM Journal of Optimization*, 11(3):796-817, 2001.

[Lau09] M. Laurent. *Sums of Squares, Moment Matrices and Optimization Over Polynomials*, pages 157-270. Springer New York, New York, NY, 2009.

[Lor05] M. Lorenz. *Multiplicative Invariant Theory*. Encyclopaedia of Mathematical Sciences. Springer, Berlin, Heidelberg, 2005.

[LSX08] H. Li, J. Sun, and Y. Xu. Discrete Fourier analysis, cubature, and interpolation on a hexagon and triangle. *SIAM Journal of Numerical Analysis*, 46(4):1653-1681, 2008.

[LSX12] H. Li, J. Sun, and Y. Xu. Discrete Fourier analysis and Chebyshev polynomials with $G_2$ group. *SIGMA. Symmetry, Integrability and Geometry. Methods and Applications*, 8:67-96, 2012.

[LX10] H. Li and Y. Xu. Discrete Fourier analysis on fundamental domain and simplex of $A_d$ lattice in $d$ variables. *The Journal of Fourier Analysis and Applications*, 16(3):383-433, 2010.

[Mac90] I. MacDonald. Orthogonal polynomials associated with root systems. In *Orthogonal Polynomials: Theory and Practice*, volume 294 of *NATO ASI Series*, pages 311-318. Springer, Netherlands, Dordrecht, 1990.

[Met22] T. Metzlaff. *Groupes Cristallographiques et Polynômes de Chebyshev en Optimisation Globale*. Thèse de doctorat en Mathématiques, dirigée par E. Hubert, Inria d'Université Côte d'Azur, theses.fr/2022COAZ4094, 2022.

[MNR13] H. Munthe-Kaas, M. Nome, and B. Ryland. Through the kaleidoscope: symmetries, groups and Chebyshev-approximations from a computational point of view. In *Foundations of computational mathematics*, Budapest 2011, London Mathematical Society Lecture Note Series, pages 188-229. Cambridge Univ. Press, Cambridge, 2013.

[MP11] R. Moody and J. Patera. Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups. *Advances in Applied Mathematics*, 47:509-535, 2011.

[Pro78] C. Procesi. Positive symmetric functions. *Advances in Mathematics*, 29(2):219-225, 1978.

[PS85] C. Procesi and G. Schwarz. Inequalities defining orbit spaces. *Inventiones mathematicae*, 81:539-554, 1985.

[Ste75] R. Steinberg. On a theorem of Pittie. *Topology*, 14(2):173-177, 1975.

[Tal20] V. Talamini. Generating formulas for finite reflection groups of the infinite series $Sn$, $An$, $Bn$ and $Dn$. *Rendiconti del Circolo Matematico di Palermo Series 2*, 69:1043-1077, 2020.

[Xu15] Y. Xu. Generalized characteristic polynomials and Gaussian cubature rules. *SIAM Journal on Matrix Analysis and Applications*, 36:1129-1142, 2015.

E. Hubert, CENTRE INRIA D'UNIVERSITÉ CÔTE D'AZUR, 06902 SOPHIA ANTIPOLIS, FRANCE

E-mail address: evelyne.hubert@inria.fr

ORCID: 0000-0003-1456-9524

T. Metzlaff, DEPARTMENT OF MATHEMATICS, RPTU KAISERSLAUTERN-LANDAU, 67663 KAISERSLAUTERN, GERMANY

Former Affiliation: CENTRE INRIA D'UNIVERSITÉ CÔTE D'AZUR, 06902 SOPHIA ANTIPOLIS, FRANCE

E-mail address: tobias.metzlaff@rptu.de

ORCID: 0000-0002-0688-7074

C. Riener, DEPARTMENT OF MATHEMATICS, UIT THE ARCTIC UNIVERSITY, 9037 TROMSØ, NORWAY

E-mail address: cordian.riener@uit.no

ORCID: 0000-0002-1192-3500