On Chevalley-Shephard-Todd's theorem in positive characteristic

TL;DR

This paper extends Chevalley-Shephard-Todd's theorem to positive characteristic, characterizing when a finite group action on a vector space is coregular based on pseudo-reflections and the direct summand property.

Contribution

It proves a Chevalley-Shephard-Todd type theorem in positive characteristic, linking coregularity to pseudo-reflections and the direct summand property for irreducible representations.

Findings

Coregular action iff generated by pseudo-reflections and has the direct summand property

Extension of classical theorem to positive characteristic fields

Characterization applies to irreducible representations

Abstract

Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants and the 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 $V$ is an irreducible $kG$-representation, then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds.