Rank functors and representation rings of quivers

To my wife Kelly. ii ACKNOWLEDGEMENTS I would like to thank, first and foremost, my advisor Harm Derksen for guiding me along the path to becoming a mathematical researcher. The many hours we spent in discussion and his creative suggestions were absolutely essential for completion of this project. I would also like to thank John Stembridge for reading this dissertation and for some helpful discussions relating to future directions; specifically, a conversation at MSRI on graph minors, and later directing me to some references on minuscule posets and representation theory. Additional thanks are due to Hugh Thomas for providing many useful comments about Chapter 3 when it was in its preprint form. He also has suggested some interesting connections to other work, as mentioned in the final section. I thank Andreas Blass for his extremely meticulous reading of the thesis which yielded a substantial improvement of the final product, and Achilleas Anastasopoulos and Farkhod Eshmatov for supporting me by being on my dissertation committee. I am indebted to the University of Michigan Department of Mathematics community in general for sustaining a stimulating environment in which to work, and preparing me for a career in mathematics. In particular, I thank Toby Stafford for his advice and support at a critical time when I was choosing an advisor and research topic. My research interests have been influenced by so many graduate students and faculty that I will not name them individually, but I will say that it has been quite an extraordinary place to become a mathematician over the past six years. Besides those that have helped me develop as a researcher, I would like to thank Gavin LaRose, Karen Rhea, and Dale Winter for helping me improve as an instructor. The work of the department staff, especially Warren Noone and Tara McQueen, is deeply appreciated for making my experience at the university a better one. I would like thank several professors at the University of Kansas for getting me interested in research while I was an undergraduate. I thank Estela Gavosto, who advised me during my four years there, helped prepare me for graduate school, and got me started on my first mathematical research project. I thank Craig Huneke for his excellent courses which drew me to algebra, and whose encouragement and enthusiasm made me understand the difference between knowing something and knowing it well. I thank Jeffrey Olafsen for giving …