# Cycle type factorizations in $GL_n F_q$

Graham Gordon, University of Washington
Wednesday, May 20, 2020 - 3:30pm to 5:00pm
Via Zoom

Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.

Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of \$\mathrm{GL}_n \mathbb{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 \mathbb{F}_q\$ of enumerating factorizations into products of regular elliptic elements. More precisely, we define a notion of cycle type for \$\mathrm{GL}_n \mathbb{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 special cases, we provide explicit formulas, using a standard character-theoretic technique due to Frobenius by introducing simplified formulas for the necessary character values. 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.

The UW combinatorics seminar is inviting you to a scheduled Zoom meeting.

Topic: UW combinatorics seminar: Graham Gordon
Time: May 20, 2020 03:30 PM Pacific Time (US and Canada)

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