Faces of generalized cluster complexes and noncrossing partitions

Abstract

Let $\Phi$ be an finite root system with corresponding reflection group $W$ and let $m$ be a nonnegative integer. We consider the generalized cluster complex ${\Delta }^m(\Phi )$ defined by S. Fomin and N. Reading and the poset $N{C}_{(m)}(W)$ of $m$-divisible noncrossing partitions defined by D. Armstrong. We give a characterization of the faces of ${\Delta }^m(\Phi )$ in terms of $N{C}_{(m)}(W)$, generalizing that of T. Brady and C. Watt given in the case $m=1$. Making use of this, we give a case free proof of a conjecture of F. Chapoton and D. Armstrong, which relates a certain refined face count of ${\Delta }^m(\Phi )$ with the Möbius function of $N{C}_{(m)}(W)$.

second version