The Cluster Complex for Cluster Poisson Varieties and Representations of Acyclic Quivers

Abstract Let X be a skew-symmetrizable cluster Poisson variety. The cluster complex ^+ (X) was introduced in 12. It codifies the theta functions on X that restrict to a character of a seed torus. Every seed s for X determines a fan realization ^+ₒ (X) of ^+ (X). For every s we provide a simple and explicit description of the cones of ^+ₒ (X) and their facets using c-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of ^+ₒ (X) in terms of F-polynomials. In case X is skew-symmetric and the quiver Q associated to s is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of ^+ₒ (X) using g-vectors of (non-necessarily rigid) objects in K^b (proj \; kQ).