Modules of covariants in modular invariant theory

TL;DR

This paper studies the structure of invariant rings under group actions using modules of covariants, generalizing classical results about freeness and generation related to reflections and subgroup structures.

Contribution

It extends Serre's classical theorem by characterizing when invariant rings are free over each other, involving the subgroup generated by reflections and invariants.

Findings

If $k[V]^H$ is free over $k[V]^G$, then $G$ is generated by $H$ and the reflections in $G$.

The invariant ring $k[V]^{Higcap W}$ is free over $k[V]^W$ and generated by invariants from $H$ and $W$.

Generalizes classical theorems to arbitrary characteristic and subgroup structures.

Abstract

Let the finite group $G$ act linearly on the vector space $V$ over the field $k$ of arbitrary characteristic. If $H<G$ is a subgroup the extension of invariant rings $k[V]^G\subset k[V]^H$ is studied using modules of covariants. An example of our results is the following. Let $W$ be the subgroup of $G$ generated by the reflections in $G$. A classical theorem due to Serre says that if $k[V]$ is a free $k[V]^G$-module then $G=W$. We generalize this result as follows. If $k[V]^H$ is a free $k[V]^G$-module then $G$ is generated by $H$ and $W$, and the invariant ring $k[V]^{H\cap W}$ is free over $k[V]^W$ and generated as an algebra by $H$-invariants and $W$-invariants.