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

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