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

Let X be a skew-symmetrizable cluster Poisson variety. The cluster complex Δ ⁺(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 Δ ⁺ₛ(X) of Δ ⁺(X). For every s we provide a simple and explicit description of the cones of Δ ⁺ₛ(X) and their facets using c-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of Δ ⁺ₛ(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 Δ ⁺ₛ(X) using g-vectors of (non-necessarily rigid) objects in Kᵇ(proj \; kQ).