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arXiv · 2407.08644

Spectrum of random-to-random shuffling in the Hecke algebra

Ilani Axelrod-Freed·Sarah Brauner·Judy Hsin-Hui Chiang·Patricia Commins·Veronica Lang
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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.

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Ilani Axelrod-Freed, Sarah Brauner, Judy Hsin-Hui Chiang, Patricia Commins, Veronica Lang. 2024-07-11. Spectrum of random-to-random shuffling in the Hecke algebra. https://arxiv.org/abs/2407.08644

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