The $q$-deformed random-to-random family in the Hecke algebra
Combinatorics
2025-11-19 v3 Rings and Algebras
Representation Theory
Abstract
We generalize Reiner--Saliola--Welker's well-known but mysterious family of *-random-to-random shuffles* from Markov chains on symmetric groups to Markov chains on the Type- Iwahori--Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in with non-negative integer coefficients. Our work generalizes work of Reiner--Saliola--Welker and Lafreni\`ere for the symmetric group, and simplifies all known proofs in this case.
Cite
@article{arxiv.2503.17580,
title = {The $q$-deformed random-to-random family in the Hecke algebra},
author = {Sarah Brauner and Patricia Commins and Darij Grinberg and Franco Saliola},
journal= {arXiv preprint arXiv:2503.17580},
year = {2025}
}
Comments
43 pages, including 4 appendices and several tables. In v3, we added Propositions 5.26 and 5.27. Comments are welcome!