Quantum Schubert cells via representation theory and ring theory

TL;DR

This paper unifies different approaches to understanding the spectra of quantum Schubert cell algebras, providing new insights, explicit models, and a fresh proof of their classification.

Contribution

It combines ring theoretic and representation theoretic methods to analyze quantum Schubert cell spectra and solves related classification and containment problems.

Findings

Unified the ring theoretic and representation theoretic approaches.

Solved the containment problem for torus invariant prime ideals.

Constructed explicit models using quantum minors.

Abstract

We resolve two questions of Cauchon and Meriaux on the spectra of the quantum Schubert cell algebras U^-[w]. The treatment of the first one unifies two very different approaches to Spec U^-[w], a ring theoretic one via deleting derivations and a representation theoretic one via Demazure modules. The outcome is that now one can combine the strengths of both methods. As an application we solve the containment problem for the Cauchon-Meriaux classification of torus invariant prime ideals of U^-[w]. Furthermore, we construct explicit models in terms of quantum minors for the Cauchon quantum affine space algebras constructed via the procedure of deleting derivations from all quantum Schubert cell algebras U^-[w]. Finally, our methods also give a new, independent proof of the Cauchon-Meriaux classification.