Pop-Stack-Sorting for Coxeter Groups

TL;DR

This paper introduces a Coxeter group generalization of the pop-stack-sorting map, proves bounds on orbit sizes related to the Coxeter number, and explores sortable elements across types with enumerative results.

Contribution

It generalizes the pop-stack-sorting map to Coxeter groups, establishes bounds on orbit sizes, and connects sortable elements in different types with enumeration results.

Findings

Maximum orbit size equals the Coxeter number for finite groups.

Bound on orbit size for compulsive maps is the Coxeter number.

Enumeration of 2-pop-stack-sortable elements in types B and A.

Abstract

Let $W$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $\mathsf{Pop}:W\to W$ to be the map that fixes the identity element and sends each nonidentity element $w$ to the meet of the elements covered by $w$ in the right weak order. When $W$ is the symmetric group $S_n$, $\mathsf{Pop}$ coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where $h$ is the Coxeter number of $W$ (with $h=\infty$ if $W$ is infinite) and $O_f(w)$ denotes the forward orbit of $w$ under a map $f$. When $W$ is finite, this result is equivalent to the statement that the maximum number of terms appearing in the Brieskorn normal form of an element of $W$ is $h-1$. More generally, we define a map $f:W\to W$ to be compulsive if for every $w\in W$, $f(w)$ is less than or equal to $\mathsf{Pop}(w)$ in the right weak order. We prove that if $f$ is compulsive, then $\sup\limits_{w\in W}|O_f(w)|\leq h$. This result is new even for symmetric groups. We prove that $2$-pop-stack-sortable elements in type $B$ are in bijection with $2$-pop-stack-sortable permutations in type $A$, which were enumerated by Pudwell and Smith. Claesson and Gudmundsson proved that for each fixed nonnegative integer $t$, the generating function that counts $t$-pop-stack-sortable permutations in type $A$ is rational; we establish analogous results in types $B$ and $\widetilde A$.