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.