Representations of the general linear groups which are irreducible over subgroups

TL;DR

This paper classifies triples of groups, representations, and subgroups where the representation remains irreducible upon restriction, advancing understanding of subgroup structures in finite classical groups.

Contribution

It provides a complete classification of triples $(G,V,H)$ with irreducible restrictions, addressing a key problem in the Aschbacher-Scott program.

Findings

Classified all triples $(G,V,H)$ with irreducible restrictions.

Identified conditions under which representations remain irreducible.

Contributed to the understanding of maximal subgroups in finite classical groups.

Abstract

We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\dar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.