Séminaire Lotharingien de Combinatoire, B66c (2011), 21 pp.

Christos A. Athanasiadis and Christina Savvidou

The Local h-Vector of the Cluster Subdivision of a Simplex

Abstract. The cluster complex \Delta(\Phi) is an abstract simplicial complex, introduced by Fomin and Zelevinsky for a finite root system \Phi. The positive part of \Delta(\Phi) naturally defines a simplicial subdivision of the simplex on the vertex set of simple roots of \Phi. The local h-vector of this subdivision, in the sense of Stanley, is computed and the corresponding \gamma-vector is shown to be nonnegative. Combinatorial interpretations to the entries of the local h-vector and the corresponding \gamma-vector are provided for the classical root systems, in terms of noncrossing partitions of types A and B. An analogous result is given for the barycentric subdivision of a simplex.

Received: October 12, 2011. Accepted: February 24, 2012. Final Version: March 15, 2012.

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