Jordan–Landau theorem for matrices over finite fields

Abstract

Given a positive integer r and a prime power q, we estimate the probability that the characteristic polynomial fA(t) of a random matrix A in GLn(Fq) is square-free with r (monic) irreducible factors when n is large. We also estimate the analogous probability that fA(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−2n−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 Fq[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×n matrices with a fixed characteristic polynomial over Fq.