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

• Authors: Christos A. Athanasiadis;Eleni Tzanaki
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2006

Let 驴 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 $${\mathcal A}^{m}_{\Phi}$$ be the arrangement of hyperplanes in V of the form $$(\alpha, x) = k$$ for $$\alpha \in \Phi$$ and $$k = 0, 1,\dots,m$$ . It is known that the number $$N^+ (\Phi, m)$$ of bounded dominant regions of $${\mathcal A}^{m}_{\Phi}$$ 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 $${\mathcal A}^{m}_{\Phi}$$ 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 驴 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 驴, 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 $${\mathcal A}^{m}_{\Phi}$$ . We also provide a dual interpretation in terms of order filters in the root poset of 驴 in the special case m = 1.