The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes

TL;DR

This paper studies the absolute order on the symmetric group, proving it is homotopy Cohen-Macaulay using constructibility concepts and computing its Euler characteristic, thus advancing understanding of Coxeter group posets.

Contribution

It introduces a constructibility framework to prove the Cohen-Macaulay property of the absolute order on symmetric groups and calculates its Euler characteristic.

Findings

Absolute order on symmetric groups is homotopy Cohen-Macaulay.

The Euler characteristic of the order complex is explicitly computed.

Answers a question posed by Reiner and the first author.

Abstract

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.