Jordan--Landau theorem for matrices over finite fields

TL;DR

This paper estimates the probability that a random matrix over a finite field has a characteristic polynomial with a specified number of irreducible factors, revealing a connection to permutation cycle structures.

Contribution

It extends Cohen's recursion method to matrices over finite fields, providing precise asymptotic probabilities for characteristic polynomial factorizations.

Findings

Probability matches permutation cycle distribution

Main term involves logarithmic factors

Error term is explicitly bounded

Abstract

Given a positive integer $r$ and a prime power $q$, we estimate the probability that the characteristic polynomial $f_{A}(t)$ of a random matrix $A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic) irreducible factors when $n$ is large. We also estimate the analogous probability that $f_{A}(t)$ has $r$ irreducible factors counting with multiplicity. In either case, the main term $(\log n)^{r-1}((r-1)!n)^{-1}$ and the error term $O((\log n)^{r-2}n^{-1})$, whose implied constant only depends on $r$ but not on $q$ nor $n$, coincide with the probability that a random permutation on $n$ letters is a product of $r$ disjoint cycles. The main ingredient of our proof is a recursion argument due to S. D. Cohen, which was previously used to estimate the probability that a random degree $n$ monic polynomial in $\mathbb{F}_{q}[t]$ is square-free with $r$ irreducible factors and the analogous probability that the polynomial has $r$ irreducible factors counting with multiplicity. We obtain our result by carefully modifying Cohen's recursion argument in the matrix setting, using Reiner's theorem that counts the number of $n \times n$ matrices with a fixed characteristic polynomial over $\mathbb{F}_{q}$.