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

Moretó, Alexander

doi:10.2140/ant.2026.20.219

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

Alexander Moretó

Vol. 20 (2026), No. 2, 219–235

DOI: 10.2140/ant.2026.20.219

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 $G$ .

Keywords

representation dimension, essential dimension, faithful representation

Mathematical Subject Classification

Primary: 20C15

Secondary: 14E07, 12F10

Milestones

Received: 25 September 2022

Revised: 10 July 2024

Accepted: 11 February 2025

Published: 16 February 2026

Authors

Alexander Moretó
Departament de Matemàtiques Universitat de València 46100 Burjassot, València Spain

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Vol. 20, No. 2, 2026