EL-Shellability and Noncrossing Partitions Associated with Well-Generated Complex Reflection Groups

TL;DR

This paper proves that the lattice of noncrossing partitions associated with well-generated complex reflection groups is EL-shellable, extending known results from real reflection groups to complex cases, and explores implications for related posets.

Contribution

It extends EL-shellability results to all well-generated complex reflection groups, including exceptional cases, and analyzes the Möbius function of related posets.

Findings

Lattice of noncrossing partitions is EL-shellable for all well-generated complex reflection groups.

EL-shellability of the poset of m-divisible noncrossing partitions is established.

Derived results on the Möbius function of these posets, confirming previous conjectures.

Abstract

In this article we prove that the lattice of noncrossing partitions is EL-shellable when associated with the well-generated complex reflection group of type $G(d,d,n)$, for $d,n\geq 3$, or with the exceptional well-generated complex reflection groups which are no real reflection groups. This result was previously established for the real reflection groups and it can be extended to the well-generated complex reflection group of type $G(d,1,n)$, for $d,n\geq 3$, as well as to three exceptional groups, namely $G_{25},G_{26}$ and $G_{32}$, using a braid group argument. We thus conclude that the lattice of noncrossing partitions of any well-generated complex reflection group is EL-shellable. Using this result and a construction by Armstrong and Thomas, we conclude further that the poset of $m$-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. Finally, we derive results on the M\"obius function of these posets previously conjectured by Armstrong, Krattenthaler and Tomie.