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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Bibliographic Details
Published in:	Discrete mathematics and theoretical computer science Vol. DMTCS Proceedings, 28th...
Main Author:	Mühle, Henri
Format:	Journal Article Conference Proceeding
Language:	English
Published:	DMTCS 22.04.2020 Discrete Mathematics & Theoretical Computer Science
Subjects:	[math.math-co]mathematics [math]/combinatorics [math.co] Combinatorics Mathematics Combinatorics
ISSN:	1365-8050, 1462-7264, 1365-8050
Online Access:	Get full text
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Summary:	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.
ISSN:	1365-8050 1462-7264 1365-8050
DOI:	10.46298/dmtcs.6363