Prime ideals in the quantum grassmannian

TL;DR

This paper studies the prime spectrum of the quantum grassmannian, providing a cell decomposition linked to Schubert cells, and shows that all primes are completely prime when the deformation parameter is not a root of unity.

Contribution

It introduces a cell decomposition of the prime spectrum of the quantum grassmannian and links it to Schubert cells, also establishing properties like normal separation and catenarity.

Findings

All primes are completely prime when q is not a root of unity.

The prime spectrum admits a cell decomposition parameterized by diagrams on partitions.

The quantum grassmannian satisfies normal separation and catenarity.

Abstract

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a torus H that acts naturally on the quantum grassmannian and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the non-negative cells in recent studies concerning the totally non-negative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity.