Faces of generalized cluster complexes and noncrossing partitions

Details Faces of generalized cluster complexes and noncrossing partitions Faces of generalized cluster complexes and noncrossing partitions

2006-05-31

arxiv.org/abs/math/0605785v2

Mathematics

Combinatorics

second version

Scientific paper

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 $NC_{(m)}(W)$ of $m$-divisible noncrossing partitions defined by D. Armstrong. We give a characterization of the faces of $\Delta^m(\Phi)$ in terms of $NC_{(m)}(W)$, 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 M\"obius function of $NC_{(m)}(W)$.

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