Bender--Knuth Billiards in Coxeter Groups

TL;DR

This work develops a unifying framework of Bender--Knuth billiards for general Coxeter groups by introducing noninvertible toggles $\tau_i$ relative to a convex set $\mathscr{L}$ and studying the induced dynamics via $\mathrm{Pro}_c=\tau_{i_n}\cdots\tau_{i_1}$ for a Coxeter element $c$. It establishes a robust futuristic/ancient dichotomy, proving that finite, right-angled, rank-3, affine types $\widetilde A$ and $\widetilde C$, and groups with complete graphs are futuristic, while many affine groups outside these families are ancient; it also gives a precise finite-case sorting result $\mathrm{Pro}_c^{\mathrm{M}(c)}(W)=\mathscr{L}$ and a method to compute $\mathrm{M}(c)$ via combinatorial AR quivers. The paper develops key tools—separators, strata, folding, and the small-root billiards graph—to characterize when billiards trajectories must eventually enter or remain in $\mathscr{L}$, and demonstrates the deep connections to classical dynamical billiards via light-beam analogies. Together, these results provide a wide, uniform panorama of when and how convex sets govern the long-term behavior of noninvertible Bender--Knuth toggles across a broad spectrum of Coxeter groups, with explicit ancient/futuristic classifications and rich directions for future exploration in combinatorics and dynamics.

Abstract

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where $I$ is a finite index set. Fix a nonempty convex subset $\mathscr{L}$ of $W$. If $W$ is of type $A$, then $\mathscr{L}$ is the set of linear extensions of a poset, and there are important Bender--Knuth involutions $\mathrm{BK}_i\colon\mathscr{L}\to\mathscr{L}$ indexed by elements of $I$. For arbitrary $W$ and for each $i\in I$, we introduce an operator $τ_i\colon W\to W$ (depending on $\mathscr{L}$) that we call a noninvertible Bender--Knuth toggle; this operator restricts to an involution on $\mathscr{L}$ that coincides with $\mathrm{BK}_i$ in type $A$. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$, we consider the operator $\mathrm{Pro}_c=τ_{i_n}\cdotsτ_{i_1}$. We say $W$ is futuristic if for every nonempty finite convex set $\mathscr{L}$, every Coxeter element $c$, and every $u\in W$, there exists an integer $K\geq 0$ such that $\mathrm{Pro}_c^K(u)\in\mathscr{L}$. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When $W$ is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of $W$, then $τ_{i_N}\cdotsτ_{i_1}(W)=\mathscr{L}$; this allows us to determine the smallest integer $\mathrm{M}(c)$ such that $\mathrm{Pro}_c^{\mathrm{M}(c)}(W)=\mathscr{L}$ for all $\mathscr{L}$. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$, $\widetilde C$, or $\widetilde G_2$.

Figures (13)

• Figure 1: The Coxeter arrangement of $\widetilde{A}_2$ forms a triangular grid whose unit triangles correspond to the elements of $\widetilde{A}_2$. The black convex set $\mathscr{L}$ turns each hyperplane in the Coxeter arrangement into either a transparent window (indicated by a thin gray line) or a one-way mirror (indicated by a line colored yellow and red). Starting at the initial unit triangle marked with the brown dot, we apply the noninvertible Bender--Knuth toggles ${\color{ArrowBlue}\tau_0},{\color{Traj5}\tau_1},{\color{Traj4}\tau_2},{\color{ArrowBlue}\tau_0},{\color{Traj5}\tau_1},{\color{Traj4}\tau_2},\ldots$. This has the effect of following a thin cyan beam of light that eventually gets trapped in $\mathscr{L}$.
• Figure 2: A stereographic projection of the Tits cone and Coxeter arrangement of $\mathfrak S_4$. The black convex set $\mathscr{L}$ turns each hyperplane in the Coxeter arrangement into either a transparent window (indicated by a thin gray circle) or a one-way mirror (indicated by a circle colored yellow and red). Regions correspond to permutations in $\mathfrak S_4$, which are represented as labelings of a $4$-element ($\mathsf{N}$-shaped) poset. Arrows indicate the billiards trajectory determined by the starting permutation $u_0=2413$ (marked with a brown dot) and the ordering ${\color{ArrowBlue}1},{\color{Traj5}2},{\color{Traj4}3}$ of $I$. This billiards trajectory follows a (thin cyan) light beam. (The diagram is not to scale, so angles have been distorted.)
• Figure 3: If we choose our convex set $\mathscr{L}$ to be an infinite strip (shown in black) in $\widetilde{A}_2$, then the billiards trajectory can "escape to infinity" without ever reflecting off of a mirror.
• Figure 4: The Tits cone and Coxeter arrangement of the Coxeter group with Coxeter graph $$ . We have passed to the positive projectivization $\mathbb{P}({\mathbb{B} W})$, which is a hyperbolic plane, and then drawn the hyperbolic plane using the Poincaré disk model. The black convex set $\mathscr{L}$ turns each hyperplane in the Coxeter arrangement into either a transparent window (indicated by a thin gray line) or a one-way mirror (indicated by a line colored yellow and red). Arrows indicate a billiards trajectory that starts at the region marked with the brown dot. The billiards trajectory follows the (thin cyan) light beam.
• Figure 5: An illustration of the proof of \ref{['lem:super-strong-acute-angle']}, drawn using the positive projectivization of the Tits cone and Coxeter arrangement of $\widetilde{G}_2$, whose Coxeter graph is $.$ The assumption that $\tau_i(u)=u$ implies that ${\mathscr{L}\subseteq H^+_{\beta}\cap H_{u^{-1}\alpha_i}^+\subseteq H_{u^{-1}\gamma}^-}$, which ends up contradicting the hypothesis that $\beta$ is a transmitting root of the stratum containing $u$.

Theorems & Definitions (112)

• Remark 1.1
• Definition 1.2
• Example 1.3
• Example 1.4
• Remark 1.5
• Definition 1.6
• Remark 1.7
• Proposition 1.7
• Definition 1.8
• Example 1.9