# ALGEBRAIC COMBINATORICS

Coxeter Pop-Tsack Torsing
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 559-581.

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 ${\mathsf{Pop}}_{T}\left(w\right)=w·{\pi }_{T}{\left(w\right)}^{-1}$, where ${\pi }_{T}\left(w\right)$ is the join in the noncrossing partition lattice $\mathrm{NC}\left(w,c\right)$ 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.

Accepted:
Published online:
DOI: 10.5802/alco.226
Classification: 05E16,  05A05,  05A18
Keywords: Coxeter group, pop-stack sorting, noncrossing partition, dual braid monoid
Defant, Colin 1; Williams, Nathan 2

1 Princeton University
2 University of Texas at Dallas
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Defant, Colin; Williams, Nathan. Coxeter Pop-Tsack Torsing. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 559-581. doi : 10.5802/alco.226. https://alco.centre-mersenne.org/articles/10.5802/alco.226/

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