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arXiv · 1511.00711

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

Joel Brewster Lewis·Alejandro H. Morales
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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.

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Joel Brewster Lewis, Alejandro H. Morales. 2015-12-16. GL_n(F_q)-analogues of factorization problems in the symmetric group. https://doi.org/10.1016/j.ejc.2016.05.004

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