Separating Invariants of Finite Groups

TL;DR

This paper extends the theory of separating invariants from linear to non-linear group actions on affine varieties, establishing conditions under which polynomial separating algebras exist and their relation to reflection groups.

Contribution

It generalizes existing results on separating invariants to non-linear actions and characterizes when polynomial separating algebras and complete intersections occur for reflection groups.

Findings

Polynomial separating algebras exist only for reflection groups.

Complete intersection separating algebras occur only for 2-reflection groups.

A separating set of size n + k - 1 exists only for k-reflection groups.

Abstract

This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac-Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the invariant ring or a separating subalgebra to properties of the group action. All these results are limited to the case of linear actions on vector spaces. The goal of this paper is to lift this restriction by extending these results to the case of (possibly) non-linear actions on affine varieties. Under mild assumptions on the variety and the group action, we prove that polynomial separating algebras can exist only for reflection groups. The benefit of this gain in generality is demonstrated by an application to the semigroup problem in multiplicative invariant theory. Then we show that separating algebras which are complete intersections in a certain codimension can exist only for 2-reflection groups. Finally we prove that a separating set of size $n + k - 1$ (where $n$ is the dimension of $X$) can exist only for $k$-reflection groups. Several examples show that most of the assumptions on the group action and the variety that we make cannot be dropped.