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SUMMARY:Shamgar Gurevich (U. Wisconsin\, Madison)
DTSTART:20210420T160000Z
DTEND:20210420T170000Z
DTSTAMP:20260423T024511Z
UID:GNTRT/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GNTRT/12/">H
 armonic Analysis on GL_n over Finite Fields</a>\nby Shamgar Gurevich (U. W
 isconsin\, Madison) as part of Geometry\, Number Theory and Representation
  Theory Seminar\n\n\nAbstract\nThere are many formulas that express intere
 sting properties of a finite group \n$G$ in terms of sums over its charact
 ers. For estimating these sums\, one of the most salient quantities to und
 erstand is the character ratio $\\mathrm{Trace}(\\rho(g))/ \\dim(\\rho)$ f
 or an irreducible representation $\\rho$ of $G$ and an element $g \\in G.$
  For example\, Diaconis and Shahshahani stated a formula of the mentioned 
 type for analyzing certain random walks on $G.$ Recently\, we discovered t
 hat for classical groups $G$ over finite fields there is a natural invaria
 nt of representations that provides strong information on the character ra
 tio. We call this invariant rank. \n\nRank suggests a new organization of 
 representations based on the very few “Small” ones. This stands in con
 trast to Harish-Chandra’s “philosophy of cusp forms”\, which is (sin
 ce the 60s) the main organization principle\, and is based on the (huge co
 llection) of “Large” representations. \n\nThis talk will discuss the n
 otion of rank for the group GLn over finite fields\, demonstrate how it co
 ntrols the character ratio\, and explain how one can apply the results to 
 verify mixing time and rate for random walks. \n\nThis is joint work with 
 Roger Howe (Yale and Texas A&M). The numerics for this work was carried wi
 th Steve Goldstein (Madison) and John Cannon (Sydney).\n
LOCATION:https://researchseminars.org/talk/GNTRT/12/
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