GL_n(F_q)-analogues of factorization problems in the symmetric group

TL;DR

This paper extends factorization counting problems from the symmetric group to the general linear group over finite fields, providing new generating functions, character-based methods, and asymptotic growth results.

Contribution

It introduces GL_n(F_q)-analogues of symmetric group factorizations, develops generating functions with attractive coefficients, and computes asymptotic growth rates for factorizations.

Findings

Generated explicit formulas for factorizations in GL_n(F_q)

Established the positivity of coefficients after basis change

Derived asymptotic growth rates for fixed genus factorizations

Abstract

We consider GL_n(F_q)-analogues of certain factorization problems in the symmetric group S_n: rather than counting factorizations of the long cycle (1, 2, ..., n) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in S_n, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. Our work generalizes several recent results on factorizations in GL_n(F_q) and also uses a character-based approach. As an application of our results, we compute the asymptotic growth rate of the number of factorizations of fixed genus of a regular elliptic element in GL_n(F_q) into two factors as n goes to infinity. We end with a number of open questions.