A numerical invariant for linear representations of finite groups

TL;DR

This paper introduces a new numerical invariant called essential dimension for linear representations of finite groups, relating it to canonical dimension and extending classical theorems, with notable differences in modular cases.

Contribution

It defines and explores the essential dimension for representations, relating it to canonical dimension and extending classical results on the Schur index.

Findings

Computed canonical dimension for a broad class of varieties

Established analogues of classical theorems for the essential dimension

Showed unexpected behaviors of essential dimension in modular settings

Abstract

We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that essential dimension of representations can behave in rather unexpected ways in the modular setting.