GIT-Equivalence and Semi-Stable Subcategories of Quiver Representations

TL;DR

This paper investigates when semi-stable subcategories of quiver representations are identical for different rational vectors, extending previous tame case results to a more general, invariant theoretic framework.

Contribution

It provides a general criterion for the equality of semi-stable subcategories in acyclic quivers, broadening the scope beyond the tame case.

Findings

Established a general criterion for GIT-equivalence of rational vectors

Extended known tame case results to all acyclic quivers

Used an invariant theoretic approach for the analysis

Abstract

In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case in [10]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.