Séminaire Lotharingien de Combinatoire, 95B.29 (2026), 12 pp.
Sarah Brauner, Patricia Commins, Darij Grinberg and Franco Saliola
The q-Deformed Random-to-Random Family in the Hecke Algebra
Abstract.
We generalize Reiner-Saliola-Welker's well-known but mysterious family of
k-random-to-random shuffles from Markov chains on symmetric groups
to Markov chains on the Type-A Iwahori-Hecke algebras. We prove that the family of
operators pairwise commutes and has eigenvalues that are polynomials in q
with non-negative integer coefficients. Our work generalizes work of
Reiner-Saliola-Welker and Lafrenière for the symmetric group, and
simplifies all known proofs in this case.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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