Cycle type factorizations in $\mathrm{GL}_n \mathbb{F}_q$

Abstract: Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of $\mathrm{GL}_n \mathbb{F}_q$ are somehow analogous to the $n$-cycles of the symmetric group. In 1981, Stanley enumerated the factorizations of permutations into products of $n$-cycles. We study the analogous problem in $\mathrm{GL}_n \mathbb{F}_q$ of enumerating factorizations into products of regular elliptic elements. More precisely, we define a notion of cycle type for $\mathrm{GL}_n \mathbb{F}_q$ and seek to enumerate the tuples of a fixed number of regular elliptic elements whose product has a given cycle type. In some special cases, we provide explicit formulas, using a standard character-theoretic technique due to Frobenius by introducing simplified formulas for the necessary character values. We also address, for large $q$, the problem of computing the probability that the product of a random tuple of regular elliptic elements has a given cycle type. We conclude with some results about the polynomiality of our enumerative formulas and some open problems.