On the minimal dimension of a faithful linear representation of a finite group

TL;DR

This paper establishes an upper bound on the minimal dimension of faithful linear representations of finite groups, with specific exceptions, and explores related bounds and connections to essential dimension.

Contribution

It proves a universal upper bound of sqrt{|G|} for the representation dimension of finite groups, except for a specific class of 2-groups, and relates this invariant to the essential dimension.

Findings

Most finite groups have a faithful linear representation of dimension at most sqrt{|G|}.

Identifies a specific class of 2-groups where the bound does not hold.

Provides bounds for representation dimensions of quotients and discusses their relation to essential dimension.

Abstract

The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a $2$-group with elementary abelian center of order $8$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ are fully ramified with respect to $G/Z(G)$. We also obtain bounds for the representation dimension of quotients of $G$ in terms of the representation dimension of $G$, and discuss the relation of this invariant with the essential dimension of $G$.