"The eigenvalues of a Markov chain determine its mixing time. In this talk, I will describe a Markov chain 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 Axelrod-Freed, Chiang, Commins and Lang, we generalize random-to-random shuffling to the (Type A) 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 considerably by drawing connections between random-to-random shuffling and the Jucys-Murphy elements of the Hecke algebra."
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