A note on gamma triangles and local gamma vectors

TL;DR

This paper introduces Gamma-triangles, demonstrating their fundamental role in cluster complex combinatorics, expressing them via local gamma-vectors, and explicitly computing these for all cluster complexes.

Contribution

It defines Gamma-triangles, relates them to local gamma-vectors, and provides explicit computations for all cluster complexes, advancing the understanding of their combinatorial structure.

Findings

Gamma-triangles are more fundamental than F- and H-triangles.

Gamma-triangles can be expressed as sums of local gamma-vectors.

Explicit formulas for Gamma-triangles in all cluster complexes.

Abstract

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.