Growth of cross-characteristic representations of finite quasisimple groups of Lie type

TL;DR

This paper establishes a bound on the number of conjugacy classes of maximal subgroups in almost simple groups with classical Lie type socles, using representation theory and character degree analysis.

Contribution

It introduces a new bound on conjugacy classes of maximal subgroups based on cross-characteristic representations and detailed character degree analysis.

Findings

Bound of $2n^{5.2}+n ext{log}_2 ext{log}_2 q$ for conjugacy classes

Derived bounds for cross-characteristic representations of simple Lie type groups

Utilized class numbers, minimal character degrees, and degree gaps

Abstract

In this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of Lie type. The bound is $2n^{5.2}+n\log_2\log_2 q$, where $n$ is the dimension of the classical socle and $q$ is the size of the defining field. To obtain the bound, we first bound the number of projective cross-characteristic representations of simple groups of Lie type as a function of the representation degree. These bounds are computed for different families of groups separately. In the computation, we use information on class numbers, minimal character degrees and gaps between character degrees.