 
Summary: THE LOCAL hVECTOR 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 hvector 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 hvector 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 hvector, barycentric subdivision, cluster
complex, cluster subdivision, vector, noncrossing partition.
1. Introduction and results
Local hvectors were introduced by Stanley [23] 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 nelement vertex set V , the local hpolynomial V (, x) is defined
