Representation Growth in positive characteristic and conjugacy classes of maximal subgroups

TL;DR

This paper investigates the growth of representations in finite groups of Lie type and symmetric groups in positive characteristic, establishing bounds on the number of representations and conjugacy classes of maximal subgroups.

Contribution

It provides new polynomial bounds on the number of representations of bounded dimension and bounds the conjugacy classes of maximal subgroups in finite almost simple groups.

Findings

Number of representations of dimension ≤ n is polynomially bounded.

Number of conjugacy classes of maximal subgroups is at most logarithmic in group size.

Results apply to symmetric, alternating, and finite groups of Lie type.

Abstract

We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the the number of representations of dimension at most n is bounded by a low degree polynomial in n. As a consequence, we show that the number of conjugacy classes of maximal subgroups of a finite almost simple group G is at most O(log|G|).