Polynomial description for the T-Orbit Spaces of Multiplicative Actions

Evelyne Hubert (1) , Tobias Metzlaff (1) , Cordian Riener (2)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
2 UiT - The Arctic University of Norway [Tromsø, Norway]

A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the complex torus. We study the structure of the associated orbit space as the image of the fundamental invariants. For the Weyl groups of types A, B, C and D, this image is a compact basic semi-algebraic set and we present the defining polynomial inequalities explicitly. We show how orbits correspond to solutions in the complex torus of symmetric polynomial systems and give a characterization of the orbit space as the positivity-locus of a symmetric real matrix polynomial. The resulting domain is the region of orthogonality for a family of generalized Chebyshev polynomials, which have connections to topics such as Fourier analysis and representations of Lie algebras.

Mots clés: Orbit Spaces; Weyl Groups; Multiplicative Invariant Theory; Semi-Algebraic Sets; Chebyshev Polynomials; Polynomial Matrix Inequalities

Domaines: Mathématiques [math]; Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]

Evelyne Hubert, Tobias Metzlaff, Cordian Riener. Polynomial description for the T-Orbit Spaces of Multiplicative Actions. 2022. ⟨hal-03590007v1⟩