Euler characteristic of the truncated order complex of generalized noncrossing partitions

TL;DR

This paper computes the Euler characteristic of the poset of generalized noncrossing partitions, extending previous work to include complex reflection groups and providing new topological insights into these combinatorial structures.

Contribution

It provides a complete calculation of the Euler characteristic for the poset of generalized noncrossing partitions, including cases for complex reflection groups, advancing the understanding of their topology.

Findings

Euler characteristic computed for the poset with extremal elements removed

Extension of results to well-generated complex reflection groups

Advancement in topological understanding of noncrossing partition posets

Abstract

The purpose of this note is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups.