A note on gamma triangles and local gamma vectors

This article introduces Gamma-triangles, which are closely related to and more fundamental than F-triangles and H-triangles that have been used in the combinatorics of cluster complexes. It is proved that Gamma-triangles can be expressed as sums of the local gamma-vectors introduced by Athanasiadis, which themselves refine the local h-vectors attached by Stanley to simplicial subdivisions. Gamma triangles of all cluster complexes are then explicitly computed.