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Harmonic analysis on GLn over finite fields

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Shamgar Gurevich
University of Wisconsin-Madison/MPIM
Die, 2019-08-20 14:00 - 15:00
MPIM Lecture Hall

There are many formulas that express interesting properties of a finite 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 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
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 certain random walks.
This is joint work with Roger Howe (Yale and Texas AM).
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