Application of Multihomogeneous Covariants to the Essential Dimension of Finite Groups

TL;DR

This paper advances the understanding of the essential dimension of finite groups by developing multihomogenization techniques, generalizing key theorems, and analyzing faithful representations to determine minimal irreducible components.

Contribution

It introduces a systematic approach to multihomogenization, generalizes existing theorems on essential dimension, and studies minimal faithful representations of finite groups.

Findings

Generalized the central extension theorem for essential dimension.

Replaced stack-involved parts of the Karpenko-Merkurjev theorem with multihomogenization.

Computed the minimal number of irreducible components for faithful representations.

Abstract

We investigate essential dimension of finite groups over arbitrary fields and give a systematic treatment of multihomogenization, introduced by H.Kraft, G.Schwarz and the author. We generalize the central extension theorem of Buhler and Reichstein and use multihomogenization to substitute and generalize the stack-involved part of the theorem of Karpenko and Merkurjev about the essential dimension of p-groups. One part of this paper is devoted to the study of completely reducible faithful representations. Amongst results concerning faithful representations of minimal dimension there is a computation of the minimal number of irreducible components needed for a faithful representation.