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
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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