Doctor of Philosophy (PhD)



Document Type



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.



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Committee Chair

Yakimov, Milen