-analogues of Factorization Problems in the Symmetric Group

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 , ź , 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 ź ∞ . We end with a number of open questions.

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