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

Abstract

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 ≥ 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 ≥ 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 mdivisible 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.

Connectivity Properties of Factorization Posets in Generated Groups

“…One of the main goals of our work is the creation of a uniform framework with which we can essentially verify both properties by the same means: a particular linear order of the chosen generating set. Such a linear order-tailored to the case of reflection groups-plays a crucial role in [2] and [31], and can indeed be seen as a precursor to one of the main definitions of this article, Definition 5.3 below.…”

“…In this section we introduce our main tool: a linear order of A that is compatible with c ∈ G. This concept is an algebraic generalization of the compatible reflection order introduced in [2], and it also appeared in [31] in the context of complex reflection groups. We recall Assumption 4.1: Red A (c) is assumed finite.…”

“…The lattice P c (W, T) of c-noncrossing W-partitions is Hurwitz-connected, and thus chainconnected. ([2,31]). Let W be a finite irreducible well-generated complex reflection group, let T be its set of reflections, and let c ∈ W be a Coxeter element.…”

Symmetric decompositions and the strong Sperner property for noncrossing partition lattices

“…The noncrossing partition lattices enjoy many nice structural properties. It is straightforward from the definition that they are graded, atomic, (locally) selfdual, and (locally) complemented, and it is a little more involved to show that they are also EL-shellable [3,9,36,42]. Another striking property is that the cardinality of NC W is given by the corresponding W-Catalan number, defined by…”

“…There are formulas for the Möbius number [2,3], and for the number of maximal chains of NC W [17,41]. The shellability of the order complex of NC W was established uniformly when W is a real reflection group [3], and case-by-case for the remaining groups [36]. Likewise, the transitivity of the Hurwitz action of the braid group on the maximal chains of NC W was shown uniformly when W is a real reflection group [21], and case-by-case for the remaining groups [6].…”