Coxeter Pop-Tsack Torsing

Abstract

Given a finite irreducible Coxeter group W with a fixed Coxeter element c, we define the Coxeter pop-tsack torsing operator Pop T : W → W by Pop T (w) = w • π T (w) −1 , where π T (w) is the join in the noncrossing partition lattice 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 Pop T and the maximum size of a forward orbit of 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 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.

Semidistrim Lattices

“…The dual pop-stack sorting operator on the lattice of order ideals of a type A root poset is equivalent to the filling operator on Dyck paths analyzed in [STT06]. The authors have defined other variants of pop-stack sorting in [DW21a,DW21b].…”

Crystal Pop-Stack Sorting and Type A Crystal Lattices

“…where h is the Coxeter number of W . Results of similar flavors were obtained for semilattice pop-stack sorting operators on ν-Tamari lattices in [Def21a] and for (some) Coxeter pop-tsack torsing operators in [DW21].…”

“…The reason for using the name "pop-stack sorting" for these operators comes from the fact that Pop Sn coincides with the original pop-stack sorting map. In our previous article [DW21], we defined and studied "dual" versions of the Coxeter pop-stack sorting operators that we called Coxeter pop-tsack torsing operators. 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.…”