Combinatorics Seminar
Wednesday, February 26, 2020 at 4:15pm to 5:15pm
Building 2, Room 139 -- MIT - Department of Mathematics, 77 Mass. Ave., Cambridge, MA
SPEAKER: Graham Gordon (University of Washington)
TITLE: Cycle type factorizations in GLn(Fq)
ABSTRACT:
Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of GLn(Fq) 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 GLn(Fq) of enumerating factorizations into products of regular elliptic elements. More precisely, we define a notion of cycle type for GLn(Fq) and seek to enumerate the tuples of a fixed number of regular elliptic elements whose product has a given cycle type. In some special cases, we provide explicit formulas, using a standard character-theoretic technique due to Frobenius. We also address, for large q, the problem of computing the probability that the product of a random tuple of regular elliptic elements has a given cycle type. We conclude with some results about the polynomiality of our enumerative formulas and some open problems.
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