The local $h$-vector of the cluster subdivision of a simplex

Details The local $h$-vector of the cluster subdivision of a simplex The local $h$-vector of the cluster subdivision of a simplex

2012-04-02

arxiv.org/abs/1204.0362v1

Seminaire Lotharingien de Combinatoire 66 (2012), Article B66c, 21pp

Mathematics

Combinatorics

21 pages, 4 figures

Scientific paper

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.

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