# Harmonic Analysis on GLn over finite fields, and Random Walks

## Department of Mathematics Colloquium

## Meeting Details

For more information about this meeting, contact Kristin Berrigan, Stephanie Zerby, Wen-Ching W. Li.

**Speaker:** Shamgar Gurevitch, University of Wisconsin

**Abstract:** : There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:
$$
trace(\rho(g))/dim(\rho),
$$
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).

## Room Reservation Information

**Room Number:** 114 McAllister

**Date:** 04/04/2019

**Time:** 3:30pm - 4:30pm