Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

Henri Mühle

Published 2015-09-23Version 1

We prove that the noncrossing partition lattices associated with the complex reflection groups $G(d,d,n)$ for $d,n\geq 2$ 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 $k\leq n$. 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 $B$.