EL-shellability and noncrossing partitions associated with well-generated complex reflection groups

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)G(d,d,n), for d,n≥3d,n≥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)G(d,1,n), for d,n≥3d,n≥3, as well as to three exceptional groups, namely G25,G26G25,G26 and G32G32, 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 mm-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.