GLn(Fq)-analogues of factorization problems in the s

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 → ∞. We end with a number of open questions.2. Regular elliptics, character theory, and the symmetric group approach 2.1. Singer cycles and regular elliptic elements. The field F q n is an n-dimensional vector space over F q , and multiplication by a fixed element in the larger field is a linear transformation.

Absolute order in general linear groups

“…Theorem confirms the special case [, Conjecture 6.3 at ]. In forthcoming work , the second author and Morales use the same techniques to confirm [, Conjecture 6.3] in full generality.…”

Cycle type factorizations inGLn𝔽q

“…As a final remark, we point out a certain incompatibility of our definition of cycle type, inspired by [18,32], with a related notion from more recent work. In [16,21,22], the dimension of the fixed space of an element in GL n F q plays the analogous role to the number of cycles in a permutation in S n . Whereas the cycle type in S n is a refinement of the number of cycles, our notion of cycle type in GL n F q is not a refinement of the dimension of the fixed space.…”

“…Recent work by Huang, Lewis, Morales, Reiner, and Stanton [16,21,22] investigated the regular elliptic elements of GL n F q , which are those matrices whose characteristic polynomial is irreducible over F q . Their work suggests that the regular elliptic elements are analogous to the n-cycles in S n from the perspective of enumerating factorizations.…”