Polynomial Separating Algebras and Reflection Groups

TL;DR

This paper extends results on polynomial separating algebras from linear representations to more general affine varieties, linking the existence of minimal separating sets to the group's reflection properties.

Contribution

It generalizes Dufresne's results to affine varieties and establishes a connection between the Cohen-Macaulay defect and the group's generation by reflections.

Findings

Minimal separating sets of invariants exist only for reflection groups under certain conditions.

The Cohen-Macaulay defect bounds the minimal number of reflections generating the group.

Extension of polynomial separating algebra results to broader algebraic group actions.

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.