Dimension-independent statistics of $\operatorname {Gl}_n(\operatorname {\mathbb {F}_q})$ via character polynomials

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.