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

Abstract

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