Total positivity is a quantum phenomenon: the grassmannian case

TL;DR

This paper reveals a deep connection between total positivity and quantum structures in the Grassmannian, showing that quantum positroids form prime ideals and providing a combinatorial and topological framework for their study.

Contribution

It establishes that quantum positroids are prime ideals in the quantum Grassmannian and offers a combinatorial description of torus-invariant prime ideals, advancing the orbit method in quantum geometry.

Findings

Quantum positroids are completely prime ideals.

Torus-invariant prime ideals are generated by quantum Plücker coordinates.

Constructed separating Ore sets for all torus-invariant primes.

Abstract

The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter $q$ is transcendental, we show that "quantum positroids" are completely prime ideals in the quantum grassmannian $A$. As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Pl\"ucker coordinates and give a combinatorial description of these generating sets. We also give a topological description of the poset of torus-invariant prime ideals in $A$, and prove a version of the orbit method for torus-invariant objects. Finally, we construct separating Ore sets for all torus-invariant primes in $A$. The latter is the first step in the Brown-Goodearl strategy to establish the orbit method for (quantum) grassmannians.