Cross-Characteristic Representations of $Sp_6(2^a)$ and Their Restrictions to Maximal Subgroups

TL;DR

This paper classifies certain irreducible modular representations of the symplectic group $Sp_6(q)$ over fields of odd characteristic when restricted to proper subgroups, advancing the understanding of subgroup structures in classical groups.

Contribution

It provides a complete classification of pairs $(V,H)$ where $V$ is an irreducible $ ext{modular}$ representation of $Sp_6(q)$ and $H$ is a proper subgroup, contributing to the Aschbacher-Scott program.

Findings

Classification of all such pairs $(V,H)$ for $Sp_6(q)$ with $q$ even.

Identification of restrictions of irreducible representations to maximal subgroups.

Enhanced understanding of subgroup structures in classical groups.

Abstract

We classify all pairs $(V,H)$, where $H$ is a proper subgroup of $G=Sp_6(q)$, $q$ even, and $V$ is an $\ell$-modular representation of $G$ for $\ell\neq 2$ which is absolutely irreducible as a representation of $H$. This problem is motivated by the Aschbacher-Scott program on classifying maximal subgroups of finite classical groups.