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 $\mathcal {T}\mathcal {S}(Q)$ 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 $\mathcal {T}\mathcal {S}(Q)$ to $(\mathbb {R}_{>0})^{Q_{0}}$ is linearly equivalent to $\mathbb {R} \times \mathbb {R}_{>0}^{Q_{1}}$.