## Mathematics Dissertations

#### Title

Orbit closures of quiver representations

Jerzy Weyman

#### Contributor(s)

Gordana Todorov, Calin Chindris, Donald King

2012

2-2012

#### Degree Grantor

Northeastern University

Ph.D.

#### Degree Name

Doctor of Philosophy

College of Science, Department of Mathematics

#### Keywords

mathematics, Dynkin quivers, geometric technique, minimal free resolutions, normal varieties, Orbit closures, representations of quivers

Mathematics

#### Abstract

Let $Q$ be a Dynkin quiver. We study orbit closures in $\mathrm{Rep}(Q, \underline{d})$, the affine space of quiver representations of a fixed dimension vector. The orbits arise from the action of $\mathrm{Gl}(\underline{d})$ on $\mathrm{Rep}(Q, \underline{d})$ and we consider their closure in the Zariski topology.

We investigate the properties of coordinate rings of orbit closures for quivers of type $A_3$ by considering the desingularization given by Reineke. We construct explicit minimal free resolutions of the defining ideals of the orbit closures thus giving us a minimal set of generators for the defining ideal. The resolution allows us to read off some geometric properties of the orbit closure. In addition, we give a characterization for the orbit closure to be Gorenstein.

Next, we investigate orbit closures of Dynkin quivers with every vertex being source or sink. We use this resolution to derive the normality of such orbit closures. As a consequence we obtain the normality of certain orbit closures of type $E$.

Finally we consider orbit closures of type equioriented $A_n$. In this context we consider varieties $Z(\beta,\gamma)$ defined by Schofield and obtain conditions for these varieties to be orbit closures. We also obtain resolutions for a class of orbit closures and recover normality for this class. This is a special case of a more general result of Abeasis, Del Fra and Kraft.

Dissertation

Kavita Sutar

#### Permanent URL

http://hdl.handle.net/2047/d20002410

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