Exterior Algebra Structure for Relative Invariants 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 $S(V^{\ast })$ 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 $S(V^{\ast })\otimes \Lambda (M^{\ast })$. This structure is then used to obtain various generalizations of usual criterions on regularity of integers.

26 pages