DOI

10.17077/etd.0zw7-ysvk

Document Type

Dissertation

Date of Degree

Spring 2019

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Kinser, Ryan

First Committee Member

Camillo, Victor

Second Committee Member

Cooper, Benjamin

Third Committee Member

Iovanov, Miodrag

Fourth Committee Member

Frohman, Charles

Abstract

We can view quiver representations of a fixed dimension vector as an algebraic variety over an algebraically closed field $K$. There is an action of the product of general linear groups on each of these varieties where the orbits of the action correspond to isomorphism classes of quiver representation. A $K$-algebra $A$ is said to have the dense orbit property if for each dimension vector, the product of the general linear group acts on each irreducible component of the module variety with a dense orbit. Under certain conditions, a $K$ algebra $A$ is representation finite if and only if it $A$ has the dense orbit property. The implication representation finite implies the dense orbit property is always true. The converse is not true in general, as shown by Chindris, Kinser, and Weyman in \cite{ryan}. Our main theorem of this thesis builds on their work to give a family of representation infinite algebras with the dense orbit property. We also give a conjectured classification of indecomposables with dense orbits. \par

In the future, we hope the work presented here can be used to find even more examples of representation infinite algebra with the dense orbit property to then develop deeper theory to classify algebras with the dense orbit property that are representation infinite.

Keywords

dense orbits, module variety, quiver representations

Pages

vi, 87 pages

Bibliography

Includes bibliographical references (pages 86-87).

Copyright

Copyright © 2019 Danny Lara

Included in

Mathematics Commons

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