Spectrum of random-to-random shuffling in the Hecke algebra
Combinatorics
2025-10-03 v3 Probability
Representation Theory
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.
Keywords
Cite
@article{arxiv.2407.08644,
title = {Spectrum of random-to-random shuffling in the Hecke algebra},
author = {Ilani Axelrod-Freed and Sarah Brauner and Judy Hsin-Hui Chiang and Patricia Commins and Veronica Lang},
journal= {arXiv preprint arXiv:2407.08644},
year = {2025}
}
Comments
Minor edits, uniform formula for certain hook eigenvalues (Corollary 1.3)