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

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

2006-05-26

arxiv.org/abs/math/0605685v1

J. Algebr. Comb. (2006) 23, 355-375

Mathematics

Combinatorics

Scientific paper

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 $m=1$.

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