Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description
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https://hdl.handle.net/10037/34443Date
2024Type
Journal articleTidsskriftartikkel
Peer reviewed
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 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 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.
Publisher
Society for Industrial and Applied MathematicsCitation
Hubert, Metzlaff, Riener. Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description. SIAM Journal on applied algebra and geometry. 2024Metadata
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Copyright 2024 The Author(s)