Recorded 18 April 2024. Sarah Brauner of the University of Quebec Montréal presents "Spectrum of random-to-random shuffling in the Hecke algebra" at IPAM's Integrability and Algebraic Combinatorics Workshop. Abstract: "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." Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-integrability-and-algebraic-combinatorics/