On the Shard Intersection Order of a Coxeter Group
Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the case of the symmetric group, we provide combinatorial models...
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| Published in | SIAM journal on discrete mathematics Vol. 27; no. 4; pp. 1880 - 1912 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Philadelphia
Society for Industrial and Applied Mathematics
01.01.2013
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0895-4801 1095-7146 |
| DOI | 10.1137/110847202 |
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| Abstract | Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the case of the symmetric group, we provide combinatorial models for shard intersections of all classical types and use this understanding to prove that the shard intersection order is EL-shellable. Further, inspired by work of Simion and Ullman on the lattice of noncrossing partitions, we show that the shard intersection order on the symmetric group admits a symmetric Boolean decomposition, i.e., a partition into disjoint Boolean algebras whose middle ranks coincide with the middle rank of the poset. Our decomposition also yields a new symmetric Boolean decomposition of the noncrossing partition lattice. [PUBLICATION ABSTRACT] |
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| AbstractList | Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the case of the symmetric group, we provide combinatorial models for shard intersections of all classical types and use this understanding to prove that the shard intersection order is EL-shellable. Further, inspired by work of Simion and Ullman on the lattice of noncrossing partitions, we show that the shard intersection order on the symmetric group admits a symmetric Boolean decomposition, i.e., a partition into disjoint Boolean algebras whose middle ranks coincide with the middle rank of the poset. Our decomposition also yields a new symmetric Boolean decomposition of the noncrossing partition lattice. [PUBLICATION ABSTRACT] |
| Author | Petersen, T. Kyle |
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| Cites_doi | 10.1007/BF01237393 10.1007/s00454-010-9243-6 10.1137/S0895480103432192 10.1016/0012-365X(91)90376-D 10.1006/aima.2000.1936 10.1007/s10801-010-0255-3 10.1016/S0012-365X(96)00365-2 10.37236/1866 10.1007/s00454-005-1171-5 10.1016/j.jcta.2011.01.001 10.37236/1459 10.1112/S0010437X09004023 10.1137/0132068 10.1016/j.aim.2007.09.002 10.2307/1999881 10.2307/27642003 10.1137/0604046 10.1016/0097-3165(76)90079-0 10.4171/dm/248 10.1090/S0002-9939-06-08534-0 |
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| References | Bancroft E. (atypb5) 1910; 2011 Brändén P. (atypb8) 2004; 11 Hersh P. (atypb15) 1999; 6 Armstrong D. (atypb2) 2009; 202 atypb19 Fomin S. (atypb11) 2007 atypb26 atypb27 atypb17 Postnikov A. (atypb22) 2008; 13 atypb18 atypb12 atypb23 atypb13 atypb24 atypb14 atypb25 atypb1 atypb3 atypb20 Wachs M. (atypb28) 2007 atypb10 atypb4 atypb7 |
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| Title | On the Shard Intersection Order of a Coxeter Group |
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