Dimension-independent statistics of $Gl_n(F_q)$ via character polynomials

TL;DR

This paper introduces size-independent statistical measures for finite general linear groups, similar to known properties of permutations, using combinatorics of categories and representation stability.

Contribution

It develops a novel approach applying combinatorics of categories to derive size-independent statistics for finite linear groups, extending ideas from permutation groups.

Findings

Expected number of small minors is size-independent

Method applies to symmetric groups and other settings

Provides new combinatorial tools for group statistics

Abstract

Picking permutations at random, the expected number of k-cycles is known to be 1/k and is, in particular, independent of the size of the permuted set. This short note gives similar size-independent statistics of finite general linear groups: ones that depend only on small minors. The proof technique uses combinatorics of categories, motivated by representation stability, and applies simultaneously to symmetric groups, finite linear groups and many other settings.