Benjamin Dequene (University of Leeds) – On the combinatorics of resolving categories for gentle trees

A gentle quiver is the data of a finite connected directed graph together with a collection of paths of length two satisfying additional conditions. A resolving subcategory of its representations is an additive subcategory that contains the projective objects and is closed by extensions and epimorphism kernels. In our framework, such a subcategory can be described combinatorially via a collection of indecomposable representations stable under some computational conditions.

In this algebraic context, a goal is to describe all resolving subcategories. To this end, we restrict ourselves to gentle trees (the directed graph is a tree) and use a geometric model to see indecomposable representations as curves on a disk. We then construct an algorithm that will enable us to compute them explicitly.

After reviewing all the essential notions and giving some motivations to understand the context, I will explain how we first describe the monogeneous resolving subcategories (generated by a single indecomposable nonprojective representation). Then, I will give some words on the design of the algorithm that allows the construction of all the resolving subcategories of any gentle tree. If time allows, I will share some expectations we can have following those results (link with tilting representations, generalization to gentle quivers, graduated cases, etc...) — all of this with combinatorial and geometrical perspectives.

This is a joint work in progress with Michael Schoonheere.