Total Positivity is a Quantum Phenomenon: The Grassmannian Case

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 q is transcendental, we show that “quantum positroids” are completely prime ideals in the quantum grassmannian O q ( G m n ( F ) ) {\mathcal O}_q(G_{mn}(\mathbb {F})) . As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Plücker 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 O q ( G m n ( F ) ) {\mathcal O}_q(G_{mn}(\mathbb {F})) , and prove a version of the orbit method for torus-invariant objects. Finally, we construct separating Ore sets for all torus-invariant primes in O q ( G m n ( F ) ) {\mathcal O}_q(G_{mn}(\mathbb {F})) . The latter is the first step in the Brown-Goodearl strategy to establish the orbit method for (quantum) grassmannians.