A representation-theoretic interpretation of positroid classes

TL;DR

This paper introduces a new algebraic framework using representation theory to interpret positroid classes, linking geometric, combinatorial, and algebraic structures in the Grassmannian.

Contribution

It defines a family of GL representations with characters representing positroid classes, providing a novel algebraic perspective and proving a conjecture of Postnikov.

Findings

Provides a new algebraic interpretation of Schubert structure constants.

Proves a conjecture relating positroid classes and Gromov-Witten invariants.

Develops an effective algorithm for decomposing positroid classes into Schubert 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.