# ALGEBRAIC COMBINATORICS

Cycle type factorizations in ${\mathrm{GL}}_{n}{𝔽}_{q}$
Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1427-1459.

Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of ${\mathrm{GL}}_{n}{𝔽}_{q}$ are somehow analogous to the $n$-cycles of the symmetric group. In 1981, Stanley enumerated the factorizations of permutations into products of $n$-cycles. We study the analogous problem in ${\mathrm{GL}}_{n}{𝔽}_{q}$ of enumerating factorizations into products of regular elliptic elements. More precisely, we define a notion of cycle type for ${\mathrm{GL}}_{n}{𝔽}_{q}$ and seek to enumerate the tuples of a fixed number of regular elliptic elements whose product has a given cycle type. In some cases, we provide explicit formulas. Our main tool is a standard character-theoretic technique due to Frobenius, which we make use of by finding simplified formulas for the necessary character values. For every case in which we are not able to compute an explicit formula, we at least determine the asymptotic behavior. We conclude with some results about the polynomiality of our enumerative formulas and some open problems.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.259
Classification: 05A15,  20C15,  11T06
Keywords: factorization enumeration, cycle type, $q$-analogues
Gordon, Graham 1

1 University of Washington Seattle, WA 98195
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Gordon, Graham. Cycle type factorizations in $\protect \mathrm{GL}_n \protect \mathbb{F}_q\protect$. Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1427-1459. doi : 10.5802/alco.259. https://alco.centre-mersenne.org/articles/10.5802/alco.259/

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