Séminaire Lotharingien de Combinatoire, 93B.138 (2025), 12 pp.
Ilani Axelrod-Freed , Sarah Brauner, Judy Hsin-Hui Chiang, Patricia Commins and Veronica Lang
Spectrum of Random-to-Random Shuffling in the Hecke Algebra
Abstract.
We generalize random-to-random shuffling from a Markov chain on the symmetric group to one on the Type A Iwahori Hecke algebra, and show that its eigenvalues are polynomials in q with non-negative integer coefficients. Setting q=1 recovers results of Dieker and Saliola, whose computation of the spectrum of random-to-random in the symmetric group resolved a nearly 20 year old conjecture by Uyemura-Reyes. Our methods simplify their proofs by drawing novel connections to the Jucys-Murphy elements of the Hecke algebra, Young seminormal forms, and the Okounkov-Vershik approach to representation theory.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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