Summary: THE LOCAL h-VECTOR OF THE CLUSTER SUBDIVISION
OF A SIMPLEX
CHRISTOS A. ATHANASIADIS AND CHRISTINA SAVVIDOU
Abstract. The cluster complex () is an abstract simplicial com-
plex, introduced by Fomin and Zelevinsky for a finite root system .
The positive part of () naturally defines a simplicial subdivision of
the simplex on the vertex set of simple roots of . The local h-vector of
this subdivision, in the sense of Stanley, is computed and the correspond-
ing -vector is shown to be nonnegative. Combinatorial interpretations
to the entries of the local h-vector and the corresponding -vector are
provided for the classical root systems, in terms of noncrossing parti-
tions of types A and B. An analogous result is given for the barycentric
subdivision of a simplex.
Key words and phrases. Local h-vector, barycentric subdivision, cluster
complex, cluster subdivision, -vector, noncrossing partition.
1. Introduction and results
Local h-vectors were introduced by Stanley  as a fundamental tool
in his theory of face enumeration for subdivisions of simplicial complexes.
Given a (finite, topological) simplicial subdivision of the abstract simplex
2V on an n-element vertex set V , the local h-polynomial V (, x) is defined