Abstract: The talk should be accessible for advanced undergraduate students.

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$(\pi(g))$ / dim$(\pi)$, 
for an irreducible representation $\pi$ 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 $GL_n$ over finite fields, illustrate how it controls the character ratio on certain elements, and explain how one can apply the results to verify mixing time and rate for related random walks.

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

Note: This will be a Zoom talk in the sense that the speaker will be attending and delivering the talk via Zoom, however the Zoom will be projected on screens in Seminar Room 1.37. Attendees may choose to view the talk in the Seminar Room or join independently via the Zoom link below.

In person attendance is available in HN 1.37. Note this is in 1.37 not the usual room.

All attendees will be asked to check in using the CBR Covid-safe Check-In app or sign in on arrival.

To join this seminar via Zoom please click here.

If you would like to join the seminar online and are not currently affiliated with ANU, please contact Martin Helmer at martin.helmer@anu.edu.au.