Coxeter Pop-Tsack Torsing

TL;DR

This paper introduces a new operator on finite Coxeter groups, generalizing pop-stack sorting, and analyzes its periodic points and orbits, revealing connections to Coxeter group types and the Coxeter number.

Contribution

It defines the Coxeter pop-tsack torsing operator and characterizes its dynamics, including periodic points and orbit sizes, for various Coxeter group types, extending previous sorting map concepts.

Findings

Identity is the unique periodic point for certain types.

Maximum orbit size equals the Coxeter number h.

Coincidental type groups have a unique orbit of size h.

Abstract

Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$, we define the Coxeter pop-tsack torsing operator $\mathsf{Pop}_T:W\to W$ by $\mathsf{Pop}_T(w)=w\cdot\pi_T(w)^{-1}$, where $\pi_T(w)$ is the join in the noncrossing partition lattice $\mathrm{NC}(w,c)$ of the set of reflections lying weakly below $w$ in the absolute order. This definition serves as a "Bessis dual" version of the first author's notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack-sorting map on symmetric groups. We show that if $W$ is coincidental or of type $D$, then the identity element of $W$ is the unique periodic point of $\mathsf{Pop}_T$ and the maximum size of a forward orbit of $\mathsf{Pop}_T$ is the Coxeter number $h$ of $W$. In each of these types, we obtain a natural lift from $W$ to the dual braid monoid of $W$. We also prove that $W$ is coincidental if and only if it has a unique forward orbit of size $h$. For arbitrary $W$, we show that the forward orbit of $c^{-1}$ under $\mathsf{Pop}_T$ has size $h$ and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.