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

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 $d,n\ge 3$, 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 $d,n\ge 3$, 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 Möbius function of these posets previously conjectured by Armstrong, Krattenthaler and Tomie.

37 pages, 4 figures. Moved the technical details of the proof of the EL-shellability of $N{C}_{G(d,d,n)}$ to the appendix. More references added

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• Non-crossing partitions of type (e,e,r)
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Cited by in corpus (5)

• Absolute order in general linear groups
• Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
• Connectivity Properties of Factorization Posets in Generated Groups
• Koszulity of dual braid monoid algebras via cluster complexes
• Real toric manifolds associated with chordal nestohedra