Abstract
We consider GLn (Fq)-analogues of certain factorization problems in the symmetric group Sn: ratherthan counting factorizations of the long cycle(1,2, . . . , n) given the number of cycles of each factor, we countfactorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis.Our work generalizes several recent results on factorizations in GLn (Fq) and also uses a character-based approach.We end with an asymptotic application and some questions.
Highlights
There is a rich vein in combinatorics of problems related to factorizations in the symmetric group Sn
There has been interest in q-analogues of such problems, replacing Sn with the general linear group GLn(Fq) over an arbitrary finite field Fq, the long cycle with a Singer cycle c, and the number of cycles with the fixed space dimension [14, 10]. We extend this approach to give the following q-analogue of Theorem 1.1
A similar phenomenon was observed in the discussion following Theorem 4.2 in [10], namely, that the counting formula qe(α)(qn − 1)k−1 for factorizations of a regular elliptic element in GLn(Fq) into k factors with fixed space codimensions given by the composition α of n is a q-analogue of the counting formula nk−1 for factorizations of an n-cycle as a genus-0 product of k cycles of specified lengths
Summary
A similar phenomenon was observed in the discussion following Theorem 4.2 in [10], namely, that the counting formula qe(α)(qn − 1)k−1 for factorizations of a regular elliptic element in GLn(Fq) into k factors with fixed space codimensions given by the composition α of n is a q-analogue of the counting formula nk−1 for factorizations of an n-cycle as a genus-0 product of k cycles of specified lengths. In GLn(Fq), the necessary character theory was worked out by Green [8] This approach has been used recently by the first-named author and coauthors to count factorizations of Singer cycles into reflections [14] and to count genus-0 factorizations (that is, those in which the codimensions of the fixed spaces of the factors sum to the codimension of the fixed space of the product) of regular elliptic elements [10]. The full-length version of this extended abstract is available as [13]
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