The $q$-deformed random-to-random family in the Hecke algebra
Sarah Brauner, Patricia Commins, Darij Grinberg, Franco Saliola
TL;DR
This work extends the classical random-to-random shuffles to the $q$-deformed Type-$A$ Iwahori–Hecke algebra setting, proving that the family $\\{\\mathcal{R}_{n,k}(q)\\}$ of operators commute and are generically diagonalizable with eigenvalues that are polynomials in $q$ with nonnegative coefficients. The authors construct a joint eigenbasis via Specht modules, index eigenvalues by horizontal strips $\\lambda\\setminus\\mu$, and provide explicit lifting recursions that relate eigenvectors across sizes. They derive closed formulas for principal eigenvalues in hook-shaped cases and prove a broad positivity result for all eigenvalues, thereby extending and simplifying prior results in the $q=1$ case and resolving a conjecture on the second-largest eigenvalue. The spectral data obtained have implications for Markov chains on Hecke algebras and reveal deeper connections between combinatorial representation theory and Hall–Littlewood–type structures, with potential applications to stochastic processes on algebraic objects. Overall, the paper unifies combinatorial and algebraic approaches to eigenvalue problems in Hecke algebras and provides a detailed framework for understanding the spectra of a broad family of symmetrized shuffling operators.
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.