Representations of extensions of simple groups

TL;DR

This paper generalizes a classical result on projective representations of finite simple groups by extending it to all low-dimensional cases and to all characteristically simple groups, broadening the understanding of their representation theory.

Contribution

It extends Feit and Tits' result to encompass all low-dimensional projective representations and all characteristically simple groups, not just simple groups.

Findings

Classifies low-dimensional projective representations of characteristically simple groups.

Identifies exceptions in the representation theory of certain groups of Lie type.

Provides a broader framework for understanding representations of complex group extensions.

Abstract

Feit and Tits (1978) proved that a nontrivial projective representation of minimal dimension of a finite extension of a finite nonabelian simple group $G$ factors through a projective representation of $G$, except for some groups of Lie type in characteristic 2; the exact exceptions for $G$ were determined by Kleidman and Liebeck (1989). We generalise this result in two ways. First we consider all low-dimensional projective representations, not just those of minimal dimension. Second we consider all characteristically simple groups, not just simple groups.