A representation-theoretic interpretation of positroid classes

Abstract

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincar\'e dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product of a Schubert polynomial and Schur polynomial, and of the 3-point Gromov-Witten invariants for Grassmannians, proving a conjecture of Postnikov. As a byproduct, we obtain an effective algorithm for decomposing positroid classes into Schubert classes.