Invariant theory of abelian transvection groups

TL;DR

This paper characterizes when an abelian group acting linearly on a vector space has a polynomial invariant ring, showing it occurs precisely when the group is generated by pseudo-reflections and the direct summand property holds.

Contribution

It proves a Chevalley--Shephard--Todd type theorem for abelian groups, linking coregularity and the direct summand property to pseudo-reflections.

Findings

Coregular action iff generated by pseudo-reflections

Direct summand property holds under the same conditions

Provides a characterization for abelian transvection groups

Abstract

Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called {\em coregular} if the invariant ring is generated by algebraically independent homogeneous invariants and the {\em direct summand property} holds if there is a surjective $k[V]^G$-linear map $\pi:k[V]\to k[V]^G$. The following Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian, then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds.