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

Lawrence H Harper,Neal Livesay,Gene B Kim

doi:10.37236/8874

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.

Highlights

In 1928, Sperner [16] proved that the Boolean order 2n has the property that it contains no antichain of cardinality larger than its largest rank level
In 1967, Rota [14] posed the following “Research Problem”: prove or disprove that the refinement order Πn shares this same property — today known as the Sperner property — for all n
The NFP was inspired by the Ford–Fulkerson theory of networks and flows, whereas the Peck property was inspired by the Hard Lefshetz Theorem from algebraic geometry

Summary

Introduction

In 1928, Sperner [16] proved that the Boolean order 2n (i.e., the poset of all subsets of {1, 2, . . . , n}) has the property that it contains no antichain (i.e., a subset of pairwise incomparable vertices) of cardinality larger than its largest rank level. Since the initial posting of our results on arXiv, Gaetz–Gao have posted a pre-print [4] containing a proof that the absolute orders of the “generalized symmetric groups” — a class of complex Coxeter groups generalizing An and Bn — are strongly Sperner [4, Corollary 3.4] Their proof was developed independently from our work in this paper, and makes use of Harper’s Product Theorem, and is based on proving the Key Facts above. If a ranked poset P has a normalized flow with respect to the counting measure, P is strongly Sperner. (νP ×Q((P × Q)k))k is log-concave, and P × Q has a normalized flow with respect to νP ×Q It follows immediately (see the proof of Corollary 7 below) from Harper’s Product Theorem that any finite product of nontrivial chains or claws is strongly Sperner.

Regular simplexes and cubes

A factorization result

Proof of Main Theorem