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

Abstract

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.