Work

Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description

Year: 2024

Type: article

Abstract: . The Weyl group of a crystallographic root system has a multiplicative 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 types [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. 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 multiplicative 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.

Source: SIAM Journal on Applied Algebra and Geometry

DOI: 10.1137/23m158173x

Authors: Evelyne Hubert, Tobias Metzlaff, Cordian Riener

Institutions: Centre Inria d'Université Côte d'Azur, AROMATH: Algebre geometrie Modelisation et Algorithmes, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, UiT The Arctic University of Norway

Language: English

Topic: Mathematical functions and polynomials

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: European Commission, Deutsche Forschungsgemeinschaft

Awards: G434337789, Polynomial Optimization, Efficiency through Moments and Algebra