GIT-Equivalence and Semi-Stable Subcategories of Quiver Representations

Tuesday, April 26, 2016 - 1:00pm
310 Strickland
Valerie Granger (Advisor: Prof. Calin Chindris)

I will begin with an introduction of the basic notions for quivers, which are simply directed graphs. Quivers and their representations occur most naturally in representation theory of finite dimensional algebras. They turn out to have an incredibly rich combinatorial and invariant-theoretic structure. In the context of quiver invariant theory, the main objects of study are the so-called semi-stable representations. These are representations that obey certain linear homogeneous inequalities coming from the celebrated Hilbert-Mumford criterion in Geometric Invariant Theory. Semi-stable representations play a central role in the study of moduli spaces of quiver representations, but they also have applications to seemingly unrelated areas. The point of departure in this thesis is the recent work of Colin Ingalls, Charles Paquette, and Hugh Thomas on the possible interactions between semi-stable subcategories for tame quivers. Their work leads to applications to the theory of cluster algebras and Artin groups. We will review the basic notions of cones and fans and apply them to our quiver set up, finally defining the necessary machinery to state the main theorem. I will also give a brief overview of results from Auslander-Reiten Theory, and state our second theorem, which recovers previous results in the case that the quiver Q is tame. Finally, I will give plenty of examples to illustrate the main results.