Compression of Finite Group Actions and Covariant Dimension, II

TL;DR

This paper advances the understanding of covariant dimension of finite groups by introducing multihomogeneous covariants, filling gaps in previous proofs, and extending earlier results with improved techniques.

Contribution

It develops new methods using multihomogeneous covariants to complete and extend prior work on covariant dimension of finite groups.

Findings

Extended covariant dimension results for finite groups.

Introduced multihomogeneous covariants as a new technique.

Filled gaps in previous proofs and simplified the arguments.

Abstract

Let $G$ be a finite group and $\phi : V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $\phi$ is faithful if $G$ acts faithfully on $\phi(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{\phi(V)}$ taken over all faithful $\phi$. In \cite{KS07} we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of \cite{KS07}, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of \cite{KS07}.