Total stability functions for type $\mathbb{A}$ quivers

Abstract

For a quiver $Q$ of Dynkin type $\mathbb{A}_n$, we give a set of $n-1$ inequalities which are necessary and sufficient for a linear stability condition (a.k.a. central charge) $Z\colon K_0(Q) \to \mathbb{C}$ to make all indecomposable representations stable. We furthermore show that these are a minimal set of inequalities defining the space $\mathcal{TS}(Q)$ of total stability conditions, considered as an open subset of $\mathbb{R}^{Q_0} \times (\mathbb{R}_{>0})^{Q_0}$. We then use these inequalities to show that each fiber of the projection of $\mathcal{TS}(Q)$ to $(\mathbb{R}_{>0})^{Q_0}$ is linearly equivalent to $\mathbb{R} \times \mathbb{R}_{>0}^{Q_1}$.