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

2019/02/22 by Harper, Lawrence H., Kim, Gene B., Livesay, Neal
#05D05 #05E99 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.08334

Abstract

Over 50 years ago, Rota posted the following celebrated `Research Problem': prove or disprove that the partial order of partitions on an n-set (i.e., the refinement order) is Sperner. A counterexample was eventually discovered by Canfield in 1978. However, Harper and Kim recently proved that a closely related order --- i.e., the refinement order on the symmetric group --- is not only Sperner, but strong Sperner. Equivalently, the well-known absolute order on the symmetric group is strong Sperner. In this paper, we extend these results by giving a concise, elegant proof that the absolute orders on the Coxeter groups A_n and B_n are strong Sperner.

Related

• The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes
• Extended weak order for the affine symmetric group
• The absolute order of a permutation representation of a Coxeter group
• Is the Symmetric Group Sperner?
• The intermediate orders of a Coxeter group