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

TL;DR

This work characterizes the nonlinear action of Weyl groups on the compact torus by giving a unified polynomial description of the real orbit space across types $\mathrm{A}_{n-1}$, $\mathrm{B}_n$, $\mathrm{C}_n$, $\mathrm{D}_n$, and $\mathrm{G}_2$. The key idea is to express the invariant ring via fundamental invariants $\theta_i$ and to encode torus-membership as a positive semidefinite Hankel (Hermite) matrix inequality $H(z)\succeq 0$, with entries built from generalized Chebyshev polynomials $\widehat{T}_{\alpha}$ in a common pattern. The authors develop a robust toolbox for symmetric polynomial systems (via Vieta relations and Hermite forms) and provide explicit closed-form matrices $H$ for each root-system type in the standard monomial basis, plus corresponding Chebyshev-based variants for the real orbit space $\mathcal{T}_{\mathbb{R}}$. These results enable direct polynomial optimization over crystallographic symmetries and connect invariant theory with practical computational tools, including a Maple package for computing the matrices $H$.

Abstract

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 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 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.

Figures (10)

• Figure 1: The real $\mathbb{T}$--orbit space for the irreducible root systems of rank $2$ and $3$.
• Figure 2: The root system$\mathrm{A}_{2}$ admits a hexagonal weight lattice $\Omega$ (circles $\circ$) with fundamental weights $\textcolor{blue}{$\omega_{1}$},\textcolor{red}{$\omega_{2}$}$ and their orbits. Here are the usual orthogonal representation with $\mathcal{W}$--symmetry (middle) and the integer representation with $\mathcal{G}$--symmetry (left). The isomorphism $\varphi:\mathbb{Z}^2\to\Omega$ yields the change of basis from $\{e_1,e_2\}$ to $\{\omega_{1},\omega_{2}\}$. The $\mathbb{T}$--orbit space of $\mathcal{G}$ (right) is the "deltoid", also known as "Steiner's hypocycloid".
• Figure 3: The root system$\mathrm{A}_{2}$ and its weight lattice in the usual orthogonal representation and the integer representation. The orbits of the fundamental weights are the blue and red lattice elements.
• Figure 4: The intersection of the semi--algebraic sets defined by the coefficients of the characteristic polynomial of $H(z)$ is the $\mathbb{T}$--orbit space associated to $\mathrm{A}_{2}$.
• Figure 5: The root system$\mathrm{C}_{2}$ and its weight lattice in the usual orthogonal representation and the integer representation. The orbits of the fundamental weights are the blue and red lattice elements.

Theorems & Definitions (62)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• Theorem 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Example 2.7
• Proposition 2.8