Polynomial Separating Algebras and Reflection Groups

Abstract

This note considers a finite algebraic group $G$ acting on an affine variety $X$ by automorphisms. Results of Dufresne on polynomial separating algebras for linear representations of $G$ are extended to this situation. For that purpose, we show that the Cohen-Macaulay defect of a certain ring is greater than or equal to the minimal number $k$ such that the group is generated by $(k+1)$-reflections. Under certain rather mild assumptions on $X$ and $G$ we deduce that a separating set of invariants of the smallest possible size $n = \dim(X)$ can exist only for reflection groups.