Given a finite irreducible Coxeter group with a fixed Coxeter element , we define the Coxeter pop-tsack torsing operator by , where is the join in the noncrossing partition lattice of the set of reflections lying weakly below 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 is coincidental or of type , then the identity element of is the unique periodic point of and the maximum size of a forward orbit of is the Coxeter number of . In each of these types, we obtain a natural lift from to the dual braid monoid of . We also prove that is coincidental if and only if it has a unique forward orbit of size . For arbitrary , we show that the forward orbit of under has size 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
Keywords: Coxeter group, pop-stack sorting, noncrossing partition, dual braid monoid
@article{ALCO_2022__5_3_559_0, author = {Defant, Colin and Williams, Nathan}, title = {Coxeter {Pop-Tsack} {Torsing}}, journal = {Algebraic Combinatorics}, pages = {559--581}, publisher = {The Combinatorics Consortium}, volume = {5}, number = {3}, year = {2022}, doi = {10.5802/alco.226}, zbl = {07555120}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.226/} }
TY - JOUR AU - Defant, Colin AU - Williams, Nathan TI - Coxeter Pop-Tsack Torsing JO - Algebraic Combinatorics PY - 2022 SP - 559 EP - 581 VL - 5 IS - 3 PB - The Combinatorics Consortium UR - http://www.numdam.org/articles/10.5802/alco.226/ DO - 10.5802/alco.226 LA - en ID - ALCO_2022__5_3_559_0 ER -
Defant, Colin; Williams, Nathan. Coxeter Pop-Tsack Torsing. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 559-581. doi : 10.5802/alco.226. http://www.numdam.org/articles/10.5802/alco.226/
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