Pop-Stack-Sorting for Coxeter Groups

Abstract

Let W be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator Pop : W → 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 Sn, Pop coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that sup w∈W |O Pop (w)| = h, where O Pop (w) is the forward orbit of w under Pop and h is the Coxeter number of W (with h = ∞ if W is infinite). More generally, we define a map f : W → W to be compulsive if for every w ∈ W , f (w) is less than or equal to Pop(w) in the right weak order. We prove that if f is compulsive, then supThis 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 Guðmundsson proved that for each fixed nonnegative integer t, the generating function that counts t-pop-stacksortable permutations in type A is rational; we establish analogous results in types B and A.

Crystal Pop-Stack Sorting and Type A Crystal Lattices

“…Every forward orbit of Pop ♦ contains v λ , which is fixed by Pop ♦ . Our main result about crystal pop-stack sorting operators extends (3), which was one of the main theorems from [Def21b].…”

“…Our main idea is to extract information about the crystal pop-stack sorting operator Pop ♦ : B λ → B λ from the information about Pop W that the first author collected in [Def21b]. The crucial tool for doing this is the key map κ : B λ → K W [Lit94].…”

“…Since K W is an order ideal of the right weak order on W and Pop W (w) ≤ R w for all w ∈ W , the map Pop W does actually restrict to a well-defined map from K W to itself. We will need the following results from [Def21b].…”

“…In all of these settings, one of the primary points of interest is the maximum size of a forward orbit of the operator in question. For example, in [Def21b], the first author proved a generalization of Ungar's theorem to an arbitrary finite irreducible Coxeter group W by showing that (3)…”

Semidistrim Lattices

“…Recently, the pop-stack sorting map has received significant attention by combinatorialists [ABH21, ABB + 19, EG21, CG19, PS19]. The first author has previously studied pop-stack sorting operators on weak orders of arbitrary Coxeter groups in [Def21b] and on ν-Tamari lattices in [Def21a]. Mühle [M 19] studied Pop ↓ L when L is congruence-uniform, where he called Pop ↓ L (x) the nucleus of x.…”