On the Sperner property for the absolute order on complex reflection groups

Christian Gaetz,Yibo Gao

doi:10.5802/alco.114

Abstract

Two partial orders on a reflection group, the codimension order and the prefix order, are together called the absolute order when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner property, except possibly for the Coxeter group of type $D_n$, for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice $NC_W$, a certain maximal interval in the absolute order, but not for the entire poset, except in the case of the symmetric group. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.

Highlights

A ranked poset P with rank decomposition P0 P1 · · · Pr is k-Sperner if no union of k antichains of P is larger than the union of the largest k ranks of P
We choose to work in the setting of general complex reflection groups, rather than just Coxeter groups
For V an n-dimensional complex vector space, a finite group W ⊂ GL(V ) is a complex reflection group of rank n if it is generated by its set of reflections T = {w ∈ W | dim(V w) = n − 1}, where V w denotes the fixed subspace of w

Summary

Introduction

A ranked poset P with rank decomposition P0 P1 · · · Pr is k-Sperner if no union of k antichains of P is larger than the union of the largest k ranks of P (see [4] for an introduction to Sperner theory). The posets N CW are known to be strongly Sperner [11, 13]; this paper follows recent work of Harper and Kim [8] in studying the problem of whether the whole absolute order Abs(W ) is strongly Sperner. We choose to work in the setting of general complex reflection groups, rather than just Coxeter groups In this generality, the two orders (the prefix and codimension orders) do not always agree. (b) Abs(W ) is strongly Sperner for any finite complex reflection group W such that the prefix order P(W ) is equal to the codimension order C(W )

Background and definitions

Counterexamples

A conjecture for type Dn