On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Abstract

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$ and coroot lattice $\check{Q}$, spanning a Euclidean space $V$. Let $m$ be a positive integer and $\aA^m_Φ$ be the arrangement of hyperplanes in $V$ of the form $(\alpha ,x)=k$ for $\alpha \in \Phi$ and $k=0,1,...,m$. It is known that the number $N^{+}(\Phi ,m)$ of bounded dominant regions of $\aA^m_Φ$ is equal to the number of facets of the positive part ${\Delta }_{+}^m(\Phi )$ of the generalized cluster complex associated to the pair $(\Phi ,m)$ by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of $\aA^m_Φ$ and conjecture that the corresponding refinement of $N^{+}(\Phi ,m)$ coincides with the $h$-vector of ${\Delta }_{+}^m(\Phi )$. We compute these refined numbers for the classical root systems as well as for all root systems when $m=1$ and verify the conjecture when $\Phi$ has type $A$, $B$ or $C$ and when $m=1$. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of $\Phi$, orbits of the action of $W$ on the quotient $\check{Q}/(mh-1)\check{Q}$ and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of $\aA^m_Φ$. We also provide a dual interpretation in terms of order filters in the root poset of $\Phi$ in the special case $m=1$.