Coxeter Pop-Tsack Torsing

Defant, Colin; Williams, Nathan

doi:10.5802/alco.226

Defant, Colin ¹; Williams, Nathan ²

¹ Princeton University
² University of Texas at Dallas

Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 559-581.

Abstract

Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$ , we define the Coxeter pop-tsack torsing operator ${Pop}_{T} : W \to W$ by ${Pop}_{T} (w) = w \cdot π_{T} {(w)}^{- 1}$ , 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.

Received: 2021-06-10
Accepted: 2022-01-12
Published online: 2022-07-07

Zbl

DOI: 10.5802/alco.226

Classification: 05E16, 05A05, 05A18
Keywords: Coxeter group, pop-stack sorting, noncrossing partition, dual braid monoid

Author's affiliations:

Defant, Colin ¹; Williams, Nathan ²

¹ Princeton University
² University of Texas at Dallas

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Defant, Colin and Williams, Nathan},
     title = {Coxeter {Pop-Tsack} {Torsing}},
     journal = {Algebraic Combinatorics},
     pages = {559--581},
     publisher = {The Combinatorics Consortium},
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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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