Markov chains on Weyl groups from the geometry of the flag variety

TL;DR

This work introduces and analyzes the Burnside process on the flag variety X = G/B for G = GL_n(\mathbb{F}_q), yielding a Markov chain on the Weyl group W via the Bruhat decomposition. It combines geometric representation theory (Springer fibers, Green polynomials) with explicit sampling algorithms to implement the process in type A, and reveals a striking large-q phenomenon: the chain clusters into buckets labeled by the Robinson–Schensted P-symbols, with transitions between buckets governed by tableaux combinatorics; for general finite Chevalley groups, the analogous buckets become Steinberg cells. The authors derive a practical algorithm for GL_n that can be implemented computationally, provide mixing-time lower bounds, and illustrate their theory with explicit GL_3(F_q) transitions and simulations in larger types. The work thus builds a bridge between stochastic processes on flag varieties and deep geometric/combinatorial structures, with potential applications to sampling problems and insights into Weyl-group Markov dynamics in general type.

Abstract

This paper studies a basic Markov chain, the Burnside process, on the space of flags $G/B$ with $G = GL_n(\mathbb{F}_q)$ and $B$ its upper triangular matrices. This gives rise to a shuffling: a Markov chain on the symmetric group realized via the Bruhat decomposition. Actually running and describing this Markov chain requires understanding Springer fibers and the Steinberg variety. The main results give a practical algorithm for all n and q and determine the limiting behavior of the chain when q is large. In describing this behavior, we find interesting connections to the combinatorics of the Robinson-Schensted correspondence and to the geometry of orbital varieties. The construction and description is then carried over to finite Chevalley groups of arbitrary type, describing a new class of Markov chains on Weyl groups.

Figures (5)

• Figure 1: The histogram for $q = 3$, $n = 4$, $t = 1000$, with the chain starting at the permutation $3~ 2~1~ 4$.
• Figure 2: The histogram for $n = 5, q = 5, t = 1000$, with the chain starting at the permutation $3~ 2~1~ 4$.
• Figure 3: The histogram for $q = 1997, n = 4, t = 1000$, starting at $3~2~1~4$. We note that $3$ elements share the same $P$-symbol with $3~2~1~4$, so the behavior here is described by Theorem \ref{['thm:limit']}.
• Figure 4: The histogram for $n = 5, q = 20011, t = 1000$, starting at $3~2~1~4~5$. We note that $4$ elements share the same $P$-symbol with $3~2~1~4~5$ in this case.
• Figure 5: The histogram for $n = 5, t = 10000, q = 20011$, starting at $3~2~1~4~5$. In this example, the chain happened to visit four different Steinberg cells, staying in each one for a certain number of steps as pictured. The four "buckets" into which these data cluster are labelled by the pictured Young tableaux according to their color.

Theorems & Definitions (47)

• Proposition 2.1
• Definition 2.2
• Proposition 2.3: see Proposition \ref{['prop:uw']}
• Example 2.4
• Proposition 2.5: see Macdonald
• Proposition 2.6
• proof
• Proposition 2.7
• Remark 2.8
• Definition 3.1