SMRI Algebra and Geometry Online, 08 October: Shamgar Gurevich – Harmonic analysis on GLₙ over finite fields

SMRI Algebra and Geometry Online Seminar
Harmonic analysis on GLₙ over finite fields
Shamgar Gurevich (University of Wisconsin, Madison)
Thursday, 08 October, 11:00–12:30 AEST (Sydney)
Other time zones: Vancouver, Wed 17:00; Toronto, Wed 20:00; London, Thu 01:00; Cape Town, Thu 02:00; Mumbai, Thu 05:30; Beijing, Thu 08:00

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations.

This talk will discuss the notion of rank for the group GLₙ over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).

Please register to attend.
The seminar will be held online via Zoom. You will be send a confirmation email with connection details.

This seminar will be recorded and made available on the SMRI Youtube channel.

Stephen Morgan