# ALGEBRAIC COMBINATORICS

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 $\mathrm{Abs}\left(W\right)$ 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 $N{C}_{W}$ [, ], a certain maximal interval in $\mathrm{Abs}\left(W\right)$, but not for the entire poset, except in the case of the symmetric group []. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.114
Classification: 20F55,  06A11,  06A07
Keywords: Absolute order, Sperner property, antichain, normalized flow, reflection group.
Gaetz, Christian 1; Gao, Yibo 1

1. Department of Mathematics Massachusetts Institute of Technology Cambridge, MA USA
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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/articles/10.5802/alco.114/

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