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The eigenvalues of a Markov chain determine its mixing time. In this talk, I will describe a Markov chain on the symmetric group called random-to-random shuffling whose eigenvalues have surprisingly elegant—though mysterious—formulas. In particular, these eigenvalues were shown to be non-negative integers by Dieker and Saliola in 2017, resolving an almost 20 year conjecture.  In recent work with Ilani Axelrod-Freed, Judy Chiang, Patricia Commins and Veronica Lang, we generalize random-to-random shuffling to a Markov chain on the Type A Iwahori Hecke algebra, and prove combinatorial expressions for its eigenvalues as a polynomial in q with non-negative integer coefficients. Our methods simplify the existing proof for q=1 by drawing novel connections between random-to-random shuffling and the Jucys-Murphy elements of the Hecke algebra

Unusual location: 2112 MSB