| | |
Summary: arXiv:math.CO/0602293v114Feb2006
h-VECTORS OF GENERALIZED ASSOCIAHEDRA AND
NONCROSSING PARTITIONS
CHRISTOS A. ATHANASIADIS, THOMAS BRADY, JON MCCAMMOND,
AND COLUM WATT
Abstract. A case-free proof is given that the entries of the h-vector of the
cluster complex (), associated by S. Fomin and A. Zelevinsky to a finite
root system , count elements of the lattice L of noncrossing partitions of
corresponding type by rank. Similar interpretations for the h-vector of the
positive part of () are provided. The proof utilizes the appearance of the
complex () in the context of the lattice L, in recent work of two of the
authors, as well as an explicit shelling of ().
1. Introduction
Let be a finite root system of rank n with corresponding finite real reflection
group W. The cluster complex () was introduced by S. Fomin and A. Zelevinsky
in the context of algebraic Y -systems [13] and cluster algebras [11, 12]. It is a pure
(n - 1)-dimensional simplicial complex which is homeomorphic to a sphere [13].
When is crystallographic there is a cluster algebra associated to and () was
realized explicitly in [9] as the boundary complex of a simplicial convex polytope,
known as a (simplicial) generalized associahedron; see [10] for an overview of cluster
|
