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 $\aA^m_\Phi$ 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 $\aA^m_\Phi$ 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 $\aA^m_\Phi$ 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 $\aA^m_\Phi$. We also provide a dual interpretation in terms of order filters in the root poset of $\Phi$ in the special case $m=1$.