Jordan--Landau theorem for matrices over finite fields

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 ${\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_q)$ 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 ${\mathbb{F}}_q$.