A strong Dixmier-Moeglin equivalence for quantum Schubert cells and an open problem for quantum Plücker coordinates

In this thesis, the algebras of primary interest are the quantum Schubert cells and the quantum Grassmannians, both of which are known to satisfy a condition on primitive ideals known as the Dixmier-Moeglin equivalence.

A stronger version of the Dixmier-Moeglin equivalence is introduced - a version which deals with all prime ideals of an algebra rather than just the primitive ideals. Quantum Schubert cells are shown to satisfy the strong Dixmier-Moeglin equivalence.

Until now, given a torus-invariant prime ideal of the quantum Grassmannian, one

could not decide which quantum Plücker coordinates it contains. Presented here is a graph-theoretic method for answering this question. This may be useful for providing a full description of the inclusions between the torus-invariant prime ideals of the quantum Grassmannian and may lead to a proof that quantum Grassmannians satisfy the strong Dixmier-Moeglin equivalence.