Total positivity is a quantum phenomenon: the grassmannian case

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 O q (G mn (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 torusinvariant prime ideals in O q (G mn (F)), and prove a version of the orbit method for torusinvariant objects. Finally, we construct separating Ore sets for all torus-invariant primes in O q (G mn (F)). The latter is the first step in the Brown-Goodearl strategy to establish the orbit method for (quantum) grassmannians.