On the Sperner property for the absolute order on complex reflection groups
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 791-800.

Two partial orders on a reflection group W, the codimension order and the prefix order, are together called the absolute order Abs(W) when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner property, except possibly for the Coxeter group of type D n , for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice NC W  [11, 13], a certain maximal interval in Abs(W), but not for the entire poset, except in the case of the symmetric group [8]. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.

Received: 2019-10-01
Revised: 2020-02-06
Accepted: 2020-02-12
Published online: 2020-06-02
DOI: https://doi.org/10.5802/alco.114
Classification: 20F55,  06A11,  06A07
Keywords: Absolute order, Sperner property, antichain, normalized flow, reflection group.
@article{ALCO_2020__3_3_791_0,
     author = {Gaetz, Christian and Gao, Yibo},
     title = {On the Sperner property for the absolute order on complex reflection groups},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {3},
     year = {2020},
     pages = {791-800},
     doi = {10.5802/alco.114},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_3_791_0/}
}
Gaetz, Christian; Gao, Yibo. On the Sperner property for the absolute order on complex reflection groups. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 791-800. doi : 10.5802/alco.114. https://alco.centre-mersenne.org/item/ALCO_2020__3_3_791_0/

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