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.