$q$-deformations of the Tsetlin library

TL;DR

The paper develops $q$-deformations of the Tsetlin library by embedding the process in the type $A$ Iwahori–Hecke algebra and linking three settings—flags, permutations, and words—via commutative inclusion/projection diagrams. It establishes explicit stationary distributions and eigenvalues for each setting, using semigroup theory on the $q$-free left regular band and lumping to derive spectra and steady states across all spaces. The main technical advance is a cohesive framework that relates Markov chains on flags to those on permutations and words, enabling exact formulas for stationary states and spectra and broad generalizations to words with repetition and coset structures. The results deepen the connection between combinatorial Markov chains, Hecke algebras, and finite geometry over finite fields, with potential applications to related stochastic processes modeled by Hecke actions. Overall, the work provides a comprehensive, algebraically grounded treatment of $q$-Tsetlin libraries and their spectral data across multiple combinatorial arenas.

Abstract

The Tsetlin library is a random shuffling process on permutations of $n$ letters, where each letter $i$ can be interpreted as a book; book $i$ is brought to the front of the bookshelf with an assigned probability $x_i$. We define a $q$-deformation of the Tsetlin library by replacing the symmetric group action on permutations by the action of the type $A$ Iwahori-Hecke algebra. We compute the stationary distribution and spectrum of this Markov chain by relating it to a Markov chain on complete flags over the finite field vector space $\mathbb{F}_q^n$ and applying techniques from semigroup theory. We also generalize the $q$-Tsetlin library to words (with repeated letters), and compute its stationary distribution and spectrum.

Figures (4)

• Figure 1: The relationship between the $q$-Tsetlin library on permutations and its specializations to $q$-random-to-top and the classical Tsetlin library.
• Figure 2: Commuting diagram relating $q$-Tsetlin libraries on flags (top), permutations (middle) and words (bottom).
• Figure 3: The poset $P_{{\sf m }}$ with ${\sf m }=(5,3,4,2)$ and with upper set ${\sf a }=(2,0,3,1)$. Nodes are labeled from bottom to top.
• Figure 4: Partial right Cayley graph for the case $q=2$ and $n=3$. On the colored arrows we abbreviate $\langle e_i\rangle$ by $\langle i\rangle$.

Theorems & Definitions (74)

• Definition 2.1: Hecke algebra
• Example 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Theorem 2.6
• Theorem 2.7
• Remark 2.8
• Example 2.9
• Example 2.10