The cluster complex for cluster Poisson varieties and representations of acyclic quivers

Abstract

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

20 pages, comments are welcome