Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime

TL;DR

This paper classifies certain irreducible quasisimple linear groups containing elements of prime order with specific irreducibility properties on subspaces, using representation theory and building on previous work.

Contribution

It provides a classification of such groups, prime orders, and field characteristics, identifying cases with fixed point subspaces of half the dimension.

Findings

Classification of groups and elements with specified irreducibility properties

Identification of cases with fixed point subspaces of dimension d/2

Extension of earlier results by DiMuro

Abstract

This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module $V=(\mathbb{F}_q)^d$, is irreducible on a subspace of the form $V(1-g)$ of dimension $d/2$. We classify $G,d,r$, the characteristic $p$ of the field $\mathbb{F}_q$, and we identify those examples where the element $g$ has a fixed point subspace of dimension $d/2$. Our proof relies on representation theory, in particular, the multiplicities of eigenvalues of $g$, and builds on earlier results of DiMuro.