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 $\mathrm{\Phi }$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 $A_F^m\mathcal{A}$^m_Phi be the arrangement of hyperplanes in $V$ of the form $(a,x)=k$ (alpha, x) = k for a Ĩ F $\alpha \in \mathrm{\Phi }$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 $A_F^m\mathcal{A}$^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 $A_F^m\mathcal{A}$^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 $\mathrm{\Phi }$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 $\mathrm{\Phi }$, 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 $A_F^m\mathcal{A}$^m_Phi . We also provide a dual interpretation in terms of order filters in the root poset of $\mathrm{\Phi }$in the special case $m=1$.

Détail

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