Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Abstract

Abstract. Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex ∆ m (Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that ∆ m (Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part ∆ m + (Φ) of ∆ m (Φ). An explicit homotopy equivalence is given between ∆ m + (Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.

Generalized noncrossing partitions and combinatorics of Coxeter groups

“…We observe that the h-vector of the complex ∆ (k) (W ) coincides with our Fuss-Narayana numbers, which follows from explicit formulas given by Fomin and Reading. We mention work of Athanasiadis and Tzanaki [11] showing that the complex ∆ Tzanaki in a uniform way [114]. Our conjecture on the dual F -triangle has also been proven by Krattenthaler [71].…”

“…Based on an earlier circulated version of this thesis, some very recent results have been obtained by Athansiadis and Tzanaki [11].…”

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

“…. ,9) (10, 10) = σ 3 σ 2 σ 1 , where σ 3 = ((1, 4, 10,7)), σ 2 = ((1, 3)) ((4, 6, 10)) ( (7,8,9)), and σ 1 = ((1, 2)) ((4, 5)), and apply this procedure, we obtain the 3-atoll in Figure 3. In the figure, the faces corresponding to cycles are shaded.…”

Reflection ordering on the groupG(m,m,n)

“…Our results are Theorems 2.1, 2.3, 2.5-2.8, 2.10 and 4.4. More precisely, Theorems 2.1 and 2.3 are concerned with the set B (m) (w) in the case when the dimension of the fixed point space of w ∈ G(m, m, n) in C n is equal to the number of disjoint cycles in the decomposition of π (w) (see ((1.2.1))): Theorem 2.1 gives the equation B (m) (w) = B (1) (w) and Theorem 2.3 computes the cardinal of the set B (m) (w) for these elements w. Theorem 2.5 reduces the study of B (m) (w) to that of two simpler sets B (m) 1 (w) and B (m) 2 (w) ( On the other hand, some algorithms for finding elements of B (m) (w) are introduced in 3.11.…”

“…For any i ∈ I 1 (w), w 1 s(i; c) is perfect for some c ∈ [m − 1] (here we use the notation in 1.2-1.3). Then ws(i; c) ∈ B (1) …”