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 Pop T :WW 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.

Received:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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