Representation growth in positive characteristic and conjugacy classes of maximal subgroups

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 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 $O\left(\right(log\left|G\right|{)}^3)$.