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

2022/03/24 by Hubert, Evelyne, Metzlaff, Tobias, Riener, Cordian
#13A50 14P10 17B22 33C52 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.13152

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.

Related

• Four Dimensional Orbit Spaces of Compact Coregular Linear Groups
• Determination of the orbit spaces of non-coregular compact linear groups with one relation among the basic polynomial invariants in the P-matrix approach
• Weyl group symmetries of the toric variety associated with Weyl chambers
• Parabolic subgroup orbits on finite root systems
• Extended affine Weyl groups and Frobenius manifolds