Orbit spaces of Weyl groups acting on compact tori: a unified and explicit polynomial description. (English) Zbl 07902819
Summary: 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.
MSC:
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 14P10 | Semialgebraic sets and related spaces |
| 17B22 | Root systems |
| 33C52 | Orthogonal polynomials and functions associated with root systems |
Keywords:
multiplicative invariants; Weyl groups; root systems; orbit spaces; semi-algebraic sets; polynomial matrix inequalities; Chebyshev polynomialsReferences:
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