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 $\mathsf{Pop}_T:W\to W$ 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.