Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Abstract

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