Unisingular representations of rank 1 finite simple groups of Lie type

TL;DR

This paper classifies all irreducible unisingular representations over complex numbers for finite simple groups of Lie type of rank 1 and certain sporadic groups, revealing their eigenvalue properties.

Contribution

It provides a complete classification of unisingular irreducible representations for specific finite simple groups, a previously unexplored area.

Findings

All complex irreducible unisingular representations of rank 1 Lie type groups identified.

Unisingular representations of almost simple sporadic groups characterized.

Eigenvalue 1 presence in all group elements' representations established.

Abstract

A representation $\Phi: G \to \mathrm{GL}_n(\mathbb{F})$ of a finite group $G$ is called unisingular if the matrix $\Phi(g)$ admits $1$ as an eigenvalue for any $g\in G$. In this paper, we determine all the complex irreducible unisingular representations of the finite simple groups of Lie type of rank $1$ and of the almost simple sporadic groups.