Dynamics of Pop-Tsack Torsing

Abstract

For a finite irreducible Coxeter group $(W,S)$ with a fixed Coxeter element $c$ and set of reflections $T$, Defant and Williams define a pop-tsack torsing operation $\mathrm{Popt}\colon W \to W$ given by $\mathrm{Popt}(w) = w \cdot π_T(w)^{-1}$ where $π_T(w) = \bigvee_{t \leq_{T}w, \ t \in T}^{NC(w,c)}t$ is the join of all reflections lying below $w$ in the absolute order in the non-crossing partition lattice $NC(w,c)$. This is a "dual" notion of the pop-stack sorting operator $\mathrm{Pops}$ introduced by Defant as a way to generalize the pop-stack sorting operator on $\mathfrak{S}_n$ to general Coxeter groups. Define the forward orbit of an element $w \in W$ to be $O_{\mathrm{Popt}}(w) = \{w, \mathrm{Popt}(w), \mathrm{Popt}^2(w), \ldots \}$. Defant and Williams established the length of the longest possible forward orbits $\max_{w \in W}|O_{\mathrm{Popt}}(w)|$ for Coxeter groups of coincidental types and type $D$ in terms of the corresponding Coxeter number of the group. In their paper, they also proposed multiple conjectures about enumerating elements with near maximal orbit length. We resolve all the conjectures that they have put forth about enumeration, and in the process we give complete classifications of these elements of Coxeter groups of types $A,B$ and $D$ with near maximal orbit lengths.