Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

TL;DR

This paper characterizes torus-invariant prime ideals in quantum matrices using totally nonnegative cells, linking algebraic properties with geometric structures, and provides explicit generators for these ideals when q is transcendental.

Contribution

It establishes a precise correspondence between torus-invariant prime ideals and totally nonnegative cells, and explicitly describes generators for these ideals in certain cases.

Findings

Quantum minors characterize torus-invariant prime ideals.

A family of minors defines a non-empty totally nonnegative cell.

Explicit generators are provided for prime ideals when q is transcendental.

Abstract

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter $q$ is transcendental over $\mathbb{Q}$.