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.

Final version, 10 pages; to appear in Israel Journal of Mathematics