Stability conditions for cyclic quivers

Let Δ_n be a cyclic quiver of n vertices and let 𝒯_n denote the category of nilpotent representations of Δ_n over a field. In the present paper, we study the stability conditions on 𝒯_n and describe the semistable subcategories of 𝒯_n with a fixed slope. This provides an alternative characterization of the thick subcategories of 𝒯_n. Further, we obtain a relationship between the Harder–Narasimhan system for Δ_n and the Ringel–Hall algebra of Δ_n.