Dimension statistics of representations of finite groups

TL;DR

This paper investigates the statistical behavior of dimensions and conjugacy class sizes in finite groups, showing they tend to be roughly constant as group parameters grow large, with new notions introduced for asymptotic constancy.

Contribution

It introduces the concepts of asymptotically constant and asymptotically log constant to describe the statistical properties of group representations and conjugacy classes in large finite groups.

Findings

Dimension data and conjugacy class sizes are statistically roughly constant in large groups.

Introduces asymptotically constant and asymptotically log constant notions.

Finds similar behavior across different classes of finite groups.

Abstract

The first part of this paper deals with unipotent and reductive groups over finite fields with $q$ elements in which either $q$ goes to infinity or $G=GL_n(q)$ and $n$ goes to infinity. The second part of the paper deals with the symmetric group $S_n$. The main conclusion that we want to bring out in the case of reductive groups $G(q)$, $q$ varying, is that the dimension data, resp. the size of conjugacy classes, is in a statistical sense, ``roughly'' constant and the same (up to taking the squares). We introduce the notion of {\it asympototically constant}, and {\it asympototically log constant} to make precise these notions, which we apply to various groups discussed in this paper including the symmetric groups $S_n$.