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

## TL;DR

This paper investigates the Sperner property in the absolute order of complex reflection groups, establishing it for certain cases and identifying exceptions, thereby advancing understanding of the combinatorial structure of these groups.

## Contribution

The paper proves the Sperner property for the absolute order on complex reflection groups where the codimension and prefix orders agree, except possibly for type D_n groups, and clarifies its status for other orders.

## Key findings

- Absolute order has the strong Sperner property in many complex reflection groups.
- The Sperner property is conjectural for Coxeter groups of type D_n.
- Neither the codimension nor the prefix order has the Sperner property in general.

## 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.