Exterior Algebra Structure for Relative Invariants of Reflection Groups

TL;DR

This paper introduces a new algebraic structure on certain components related to reflection groups, enabling generalizations of criteria for the regularity of integers in invariant theory.

Contribution

It defines an algebra structure on isotypic components of tensor products involving coordinate rings and exterior algebras for reflection groups, extending classical regularity criteria.

Findings

New algebra structure on isotypic components

Generalizations of regularity criteria for integers

Applications to invariant theory of reflection groups

Abstract

Let $G$ be a reflection group acting on a vector space $V$ (over a field with zero characteristic). We denote by $S(V^*)$ the coordinate ring of $V$, by $M$ a finite dimensional $G$-module and by $\chi$ a one-dimensional character of $G$. In this article, we define an algebra structure on the isotypic component associated to $\chi$ of the algebra $S(V^*) \otimes \Lambda(M^*)$. This structure is then used to obtain various generalizations of usual criterions on regularity of integers.