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

TL;DR

This paper studies the structure of the cluster complex for skew-symmetrizable cluster Poisson varieties, providing explicit descriptions of cones and theta functions, especially for acyclic quivers, linking them to representation theory.

Contribution

It offers explicit descriptions of the cones in the cluster complex using c-vectors and formulas for theta functions via F-polynomials, connecting cluster algebra and representation theory.

Findings

Explicit cone descriptions using c-vectors.

Formulas for theta functions in terms of F-polynomials.

Connection between hyperplanes of cones and g-vectors in derived categories.

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 ${ \bf s}$ for $\mathcal{X}$ determines a fan realization $\Delta^+_{\bf s}(\mathcal{X})$ of $\Delta^+(\mathcal{X})$. For every ${\bf s}$ we provide a simple and explicit description of the cones of $\Delta^+_{{\bf s}}(\mathcal{X})$ and their facets using ${\bf c}$-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of $\Delta^+_{{ \bf s}}(\mathcal{X})$ in terms of $F$-polynomials. In case $\mathcal{X}$ is skew-symmetric and the quiver $Q$ associated to ${\bf s}$ is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of $\Delta^+_{\bf s}(\mathcal{X})$ using ${\bf g}$-vectors of (non-necessarily rigid) objects in $\mathsf{K}^{\rm b}(\text{proj} \; kQ)$.