Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Abstract

Let $Φ$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $Δ^m (Φ)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $Δ^m (Φ)$ 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 $Δ^m_+ (Φ)$ of $Δ^m (Φ)$. An explicit homotopy equivalence is given between $Δ^m_+ (Φ)$ and the poset of generalized noncrossing partitions, associated to the pair $(Φ, m)$ by D. Armstrong.