Speaker: Shamgar Gurevich
Date: October 01, 2020
Affiliation: University of Wisconsin, USA

Abstract: 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$(\rho(g)) / \dim(\rho)$, for an irreducible representation $\rho$ of $G$ and an element $g$ of $G$. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on $G$. <p> 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. <p> 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. <p> This talk will discuss the notion of rank for the group $GL_n$ 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. <p> 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).