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

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

TL;DR

This work extends random-to-random shuffling from the symmetric group to the Type A Iwahori Hecke algebra ${\mathcal{H}}_n(q)$ and derives the full eigen-spectrum of the associated operator ${\mathcal{R}}_n(q)$. The authors develop a novel approach that blends the Okounkov–Vershik representation framework with Jucys–Murphy elements, Young idempotents, and seminormal bases to produce a recursion that constructs eigenvectors across sizes. The spectrum is shown to be a family of polynomials in $q$ with nonnegative coefficients, indexed by horizontal strips ${\lambda\setminus\mu}$ with $|\lambda|=n$, and multiplicities are governed by $f^{\lambda}$ and desarrangement counts $d^{\mu}$. The analysis reveals a Deep link between random walks on ${\mathcal{H}}_n(q)$ and flag-variety dynamics over finite fields, connects to the derangement representation, and yields explicit special cases (e.g., hook shapes) alongside the Mallows stationary distribution. Overall, the paper unifies representation-theoretic techniques with probabilistic shuffling models and provides a concrete, positive-parameter spectral description of a fundamental Markov operator on Hecke algebras, with potential implications for related stochastic processes and quantum algebra contexts.

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.

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

TL;DR

This work extends random-to-random shuffling from the symmetric group to the Type A Iwahori Hecke algebra and derives the full eigen-spectrum of the associated operator . The authors develop a novel approach that blends the Okounkov–Vershik representation framework with Jucys–Murphy elements, Young idempotents, and seminormal bases to produce a recursion that constructs eigenvectors across sizes. The spectrum is shown to be a family of polynomials in with nonnegative coefficients, indexed by horizontal strips with , and multiplicities are governed by and desarrangement counts . The analysis reveals a Deep link between random walks on and flag-variety dynamics over finite fields, connects to the derangement representation, and yields explicit special cases (e.g., hook shapes) alongside the Mallows stationary distribution. Overall, the paper unifies representation-theoretic techniques with probabilistic shuffling models and provides a concrete, positive-parameter spectral description of a fundamental Markov operator on Hecke algebras, with potential implications for related stochastic processes and quantum algebra contexts.

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.
Paper Structure (32 sections, 47 theorems, 276 equations)

This paper contains 32 sections, 47 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. (Semisimple) Representation theory of $\mathcal{H}_n(q)$
  3. Basics of the Hecke algebra and conventions
  4. Explicit constructions of Specht modules using the Jucys-Murphy elements
  5. Young's idempotents
  6. Specht modules and the seminormal basis
  7. Branching rules for $\mathcal{H}_n(q)$ and the Horizontal Strip Lemma
  8. Branching rules and the Frobenius characteristic map
  9. Frobenius characteristic map
  10. The operators $\Phi$
  11. The horizontal strip lemma
  12. Flags of $\mathbb{F}_q^n$ and the spectrum of ${\mathcal{B}}_n^*(q)$
  13. The Derangement representation
  14. Connection between $\mathcal{H}_n(q)$ and flags over a finite field
  15. Random-to-top on flags
  16. ...and 17 more sections

Key Result

Theorem 1.2

For any $q \in {\mathbb{C}},$ the right action of ${\mathcal{R}}_n(q)$ on $\mathcal{H}_n(q)$ has the following properties:

Theorems & Definitions (119)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Remark 2.2: Anti-isomorphism $*$
  • Remark 2.3: Bottom versus Top
  • Corollary 2.4
  • Lemma 2.5
  • ...and 109 more