Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which as...
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| Vydané v: | Discrete mathematics and theoretical computer science Ročník DMTCS Proceedings, 28th... |
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| Hlavný autor: | |
| Médium: | Journal Article Konferenčný príspevok.. |
| Jazyk: | English |
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DMTCS
22.04.2020
Discrete Mathematics & Theoretical Computer Science |
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| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
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| Shrnutí: | We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B. |
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| ISSN: | 1365-8050 1462-7264 1365-8050 |
| DOI: | 10.46298/dmtcs.6363 |