Factorization in Unitary Loop Groups and Reduced Words in Affine Weyl Groups

As members of the Dissertation Committee, we certify that we have read the disser-tation prepared by Benjamin Pittman-Polletta entitled Factorization in Unitary Loop Groups and Reduced Words in Affine Weyl Groups and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Dedication For my family, the largest constant positive force in my life. 5 Acknowledgments My advisor, Doug Pickrell, has been a steadfast advocate and a patient supporter, guiding me with wisdom and honesty on an idiosyncratic and often rocky path to mathematical independence. His toughness, dedication, and power as a mathematician never cease to amaze me. As a teacher, the breadth and depth of his knowledge, and his unique perspective on the bigger picture, have been an inspiration. I was lucky to have him. Most of the members of my dissertation committee-were also on the committee for my comprehensive exam. All of them have been sources of advice, encouragement, feedback, and interesting mathematical discussion for years. My many mathematics teachers took pains to initiate me into the beautiful secrets of this sometimes thorny subject, even when I consigned it to the dust bin. I couldn't have asked for a better companion on the long and arduous trek into Lie theory than McKenzie Lamb. Tom LaGatta inspired, prodded, aided, annoyed, and entertained me during our long quest to become mathematicians. and many others contributed invaluable …