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

Athanasiadis, Christos A. ; Tzanaki, Eleni

Journal of Algebraic Combinatorics, Tome 23 (2006) no. 4, pp. 355-375.

Résumé

Summary: Let $\Phi $be an irreducible crystallographic root system with Weyl group $W$ and coroot lattice $\check Q$ checkQ , spanning a Euclidean space $V$. Let $m$ be a positive integer and $A ^{ m} _{ F} {\mathcal A}$^m_Phi be the arrangement of hyperplanes in $V$ of the form $( a, x) = k$ (alpha, x) = k for a Ĩ F $\alpha \in \Phi $and $k = 0, 1,\dots , m$ k = 0, 1,…,m . It is known that the number $N ^{+} ( F, m)$ N^+ (Phi, m) of bounded dominant regions of $A ^{ m} _{ F} {\mathcal A}$^m_Phi is equal to the number of facets of the positive part D $^{ m} _{+}$ ( F) Delta^m_+ (Phi) of the generalized cluster complex associated to the pair $( F, m)$ (Phi, m) by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of $A ^{ m} _{ F} {\mathcal A}$^m_Phi and conjecture that the corresponding refinement of $N ^{+} ( F, m)$ N^+ (Phi, m) coincides with the $h$ h-vector of D $^{ m} _{+}$ ( F) 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 $\check Q / ( mh -1) \check Q$ checkQ /   (mh-1)   checkQ 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 $A ^{ m} _{ F} {\mathcal A}$^m_Phi . We also provide a dual interpretation in terms of order filters in the root poset of $\Phi $in the special case $m = 1$.

Détail

Classification : 20F55;, Secondary-05E99,, 20H15, C
Keywords: keywords Catalan arrangement, bounded region, generalized cluster complex, positive part, $h$-vector