The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

There are several classes of ranked posets related to reflection groups which are known to have the Sperner property, including the Bruhat orders and the generalized noncrossing partition lattices (i.e., the maximal intervals in absolute orders). In 2019, Harper–Kim proved that the absolute orders on the symmetric groups are (strongly) Sperner. In this paper, we give an alternate proof that extends to the signed symmetric groups and the dihedral groups. Our simple proof uses techniques inspired by Ford–Fulkerson's theory of networks and flows, and a product theorem.

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

International audience 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.