Doctor of Philosophy (PhD)
In recent years the quantum Schubert cell algebras, introduced by Lusztig and De Concini--Kac, and Procesi, have garnered much interest as this versatile class of objects are furtive testing grounds for noncommutative algebraic geometry. We unify the two main approaches to analyzing the structure of the torus-invariant prime spectra of quantum Schubert cell algebras, a ring theoretic one via Cauchon's deleting derivations and a representation theoretic characterization of Yakimov via Demazure modules. As a result one can combine the strengths of the two approaches. In unifying the theories, we resolve two questions of Cauchon and Mériaux, one of which involves the Cauchon diagram containment problem. Moreover, we discover explicit quantum-minor formulas for the final generators arising from iterating the deleting derivation method on any quantum Schubert cell algebras. These formulas will play a large role in subsequent research. Lastly, we provide an independent and elegant proof of the Cauchon--Mériaux classification. The main results in this thesis appear in arXiv:1203.3780 and are joint with Milen Yakimov.
Document Availability at the Time of Submission
Secure the entire work for patent and/or proprietary purposes for a period of one year. Student has submitted appropriate documentation which states: During this period the copyright owner also agrees not to exercise her/his ownership rights, including public use in works, without prior authorization from LSU. At the end of the one year period, either we or LSU may request an automatic extension for one additional year. At the end of the one year secure period (or its extension, if such is requested), the work will be released for access worldwide.
Geiger, Joel Benjamin, "The Ring Theory and the Representation Theory of Quantum Schubert Cells" (2013). LSU Doctoral Dissertations. 1967.