ALGEBRAIC COMBINATORICS

no. 3
On the Sperner property for the absolute order on complex reflection groups
Gaetz, Christian1; Gao, Yibo1
1 Department of Mathematics Massachusetts Institute of Technology Cambridge, MA USA
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  [, ], a certain maximal interval in Abs(W), 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.

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

[1] Armstrong, Drew Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc., Volume 202 (2009) no. 949, p. x+159 | DOI | MR | Zbl

[2] Brady, Thomas; Watt, Colum K(π,1)’s for Artin groups of finite type, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), Volume 94 (2002), pp. 225-250 | DOI | MR | Zbl

[3] Carter, Roger W. Conjugacy classes in the Weyl group, Compositio Math., Volume 25 (1972), pp. 1-59 | Numdam | MR | Zbl

[4] Engel, Konrad Sperner theory, Encyclopedia of Mathematics and its Applications, 65, Cambridge University Press, Cambridge, 1997, x+417 pages | DOI | MR | Zbl

[5] Foster-Greenwood, Briana Comparing codimension and absolute length in complex reflection groups, Comm. Algebra, Volume 42 (2014) no. 10, pp. 4350-4365 | DOI | MR | Zbl

[6] Gaetz, Christian; Gao, Yibo A combinatorial 𝔰𝔩 2 -action and the Sperner property for the weak order, Proc. Amer. Math. Soc., Volume 148 (2020) no. 1, pp. 1-7 | DOI | MR | Zbl

[7] Harper, Lawrence H. The morphology of partially ordered sets, J. Combinatorial Theory Ser. A, Volume 17 (1974), pp. 44-58 | DOI | MR | Zbl

[8] Harper, Lawrence H.; Kim, Gene B. Is the Symmetric Group Sperner? (2019) (https://arxiv.org/abs/1901.00197)

[9] Harper, Lawrence H.; Kim, Gene B.; Livesay, Neal The Absolute Orders on the Coxeter Groups A n and B n are Sperner (2019) (https://arxiv.org/abs/1902.08334) | Zbl

[10] Huang, Jia; Lewis, Joel Brewster; Reiner, Victor Absolute order in general linear groups, J. Lond. Math. Soc. (2), Volume 95 (2017) no. 1, pp. 223-247 | DOI | MR | Zbl

[11] Mühle, Henri Symmetric decompositions and the strong Sperner property for noncrossing partition lattices, J. Algebraic Combin., Volume 45 (2017) no. 3, pp. 745-775 | DOI | MR | Zbl

[12] Proctor, Robert A.; Saks, Michael E.; Sturtevant, Dean G. Product partial orders with the Sperner property, Discrete Math., Volume 30 (1980) no. 2, pp. 173-180 | DOI | MR | Zbl

[13] Reiner, Victor Non-crossing partitions for classical reflection groups, Discrete Math., Volume 177 (1997) no. 1-3, pp. 195-222 | DOI | MR | Zbl

[14] Shephard, Geoffrey C.; Todd, J. A. Finite unitary reflection groups, Canadian J. Math., Volume 6 (1954), pp. 274-304 | DOI | MR | Zbl

[15] Shi, Jian-yi Formula for the reflection length of elements in the group G(m,p,n), J. Algebra, Volume 316 (2007) no. 1, pp. 284-296 | DOI | MR | Zbl

[16] Solomon, Louis Invariants of finite reflection groups, Nagoya Math. J., Volume 22 (1963), pp. 57-64 | DOI | MR | Zbl

[17] Stanley, Richard P. Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods, Volume 1 (1980) no. 2, pp. 168-184 | DOI | MR | Zbl

[18] The Sage Developers SageMath, the Sage Mathematics Software System (Version 8.9), 2019 (http://www.sagemath.org/)

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