Asymptotics for the Enumeration of Commuting Matrices over Finite Fields

TL;DR

This work analyzes the asymptotic enumeration of commuting $n\times n$ matrices over the finite field $\mathbb{F}_{p^r}$ by exploiting Feit–Fine's product-form generating function. The authors derive a precise asymptotic expansion for $Q_{p^r}(n)$ with a dominant term $p^{r(n^2+n)} \prod_{j\ge1} (1-p^{-r j})^{-j}$ and a structured correction given by coefficients $C_{m,p^r}(n)$ defined via $P_{m,p^r}$ and $F_{p^r}$; an explicit bound on the error is provided and the expansion is shown to be effective when truncated. They connect these results to Cohen–Lenstra series and analyze nilpotent classes, obtaining a convergent series representation for the nilpotent-coefficient sequence and offering numerical illustrations in low-characteristic cases. The paper concludes with open problems, including potential recursive structures, combinatorial interpretations, sharper coefficient bounds, and extensions to multivariable product identities, highlighting the broad relevance to both combinatorics and arithmetic geometry.

Abstract

We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.

Theorems & Definitions (11)

• Theorem 1.1: Feit--Fine, FF
• Theorem 1.2
• Remark
• Example
• Proposition 2.1
• Proposition 2.2: Wang, Wang
• Lemma 3.1
• Corollary 3.2
• Proposition 3.3
• Theorem 4.1