Two partial orders on a reflection group , the codimension order and the prefix order, are together called the absolute order 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 , for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice [, ], a certain maximal interval in , 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
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}, pages = {791--800}, publisher = {MathOA foundation}, volume = {3}, number = {3}, year = {2020}, doi = {10.5802/alco.114}, mrnumber = {4113607}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.114/} }
TY - JOUR AU - Gaetz, Christian AU - Gao, Yibo TI - On the Sperner property for the absolute order on complex reflection groups JO - Algebraic Combinatorics PY - 2020 SP - 791 EP - 800 VL - 3 IS - 3 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.114/ DO - 10.5802/alco.114 LA - en ID - ALCO_2020__3_3_791_0 ER -
%0 Journal Article %A Gaetz, Christian %A Gao, Yibo %T On the Sperner property for the absolute order on complex reflection groups %J Algebraic Combinatorics %D 2020 %P 791-800 %V 3 %N 3 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.114/ %R 10.5802/alco.114 %G en %F 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. http://www.numdam.org/articles/10.5802/alco.114/
[1] Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc., Volume 202 (2009) no. 949, p. x+159 | DOI | MR | Zbl
[2] ’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] Conjugacy classes in the Weyl group, Compositio Math., Volume 25 (1972), pp. 1-59 | Numdam | MR | Zbl
[4] Sperner theory, Encyclopedia of Mathematics and its Applications, 65, Cambridge University Press, Cambridge, 1997, x+417 pages | DOI | MR | Zbl
[5] Comparing codimension and absolute length in complex reflection groups, Comm. Algebra, Volume 42 (2014) no. 10, pp. 4350-4365 | DOI | MR | Zbl
[6] A combinatorial -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] The morphology of partially ordered sets, J. Combinatorial Theory Ser. A, Volume 17 (1974), pp. 44-58 | DOI | MR | Zbl
[8] Is the Symmetric Group Sperner? (2019) (https://arxiv.org/abs/1901.00197)
[9] The Absolute Orders on the Coxeter Groups and are Sperner (2019) (https://arxiv.org/abs/1902.08334)
[10] Absolute order in general linear groups, J. Lond. Math. Soc. (2), Volume 95 (2017) no. 1, pp. 223-247 | DOI | MR | Zbl
[11] 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] Product partial orders with the Sperner property, Discrete Math., Volume 30 (1980) no. 2, pp. 173-180 | DOI | MR | Zbl
[13] Non-crossing partitions for classical reflection groups, Discrete Math., Volume 177 (1997) no. 1-3, pp. 195-222 | DOI | MR | Zbl
[14] Finite unitary reflection groups, Canadian J. Math., Volume 6 (1954), pp. 274-304 | DOI | MR | Zbl
[15] Formula for the reflection length of elements in the group , J. Algebra, Volume 316 (2007) no. 1, pp. 284-296 | DOI | MR | Zbl
[16] Invariants of finite reflection groups, Nagoya Math. J., Volume 22 (1963), pp. 57-64 | DOI | MR | Zbl
[17] 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] SageMath, the Sage Mathematics Software System (Version 8.9), 2019 (http://www.sagemath.org/)
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