Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
Henri Mu ̈hle  1  
1 : Laboratoire dínformatique de l\'École polytechnique [Palaiseau]  (LIX)  -  Website
Ecole Polytechnique, Centre National de la Recherche Scientifique : UMR7161
Route de Saclay 91128 PALAISEAU CEDEX -  France

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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