Derived Quiver Representations: Reflections, Stability and Filtrations
October 2021
My Honours thesis, supervised by Asilata Bapat. We study the bounded derived category of quiver representations for an acyclic quiver Q. This can be described very concretely because the abelian category of Q-representations is hereditary. We then use reflection functors to construct explicit equivalences between the derived categories of quivers with the same underlying graph, but whose representation categories are distinct. Passing to the derived category thus makes the representation theory of quivers whose underlying graphs are acyclic entirely uniform under changes of orientation.
The final chapter discusses the iterated weight filtration, which can be computed for any artinian lattice, given a weight function. This gives a refinement of the Harder–Narasimhan filtration coming from a stability condition on the derived category. The refinement depends only on a choice of positive weights at each vertex of Q, independent of the orientation.