Equivariant Schubert calculus and geometric Satake

TL;DR

This paper uses the geometric Satake correspondence to explain the classical Schubert calculus multiplication rules, extending to equivariant cases and clarifying the link with symmetric functions and double Schur polynomials.

Contribution

It provides a new geometric and representation-theoretic explanation for Schubert calculus rules, including equivariant cases, connecting cohomology of Grassmannians with symmetric functions.

Findings

Reveals the representation-theoretic structure underlying Schubert calculus.

Extends the classical results to equivariant cohomology and double Schur polynomials.

Provides a geometric Satake-based proof of multiplication rules in cohomology.

Abstract

The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian $G(n,m)$ are the same as multiplication rules for the basis of Schur polynomials in the ring of symmetric polynomials. In this paper, we explain how to recover this somewhat mysterious connection by using the geometric Satake correspondence to put the structure of a representation of $GL_m$ on $H^\bullet(G(n,m))$ and comparing it to the Fock space representation on symmetric polynomials. This proof also extends to equivariant Schubert calculus, and gives an explanation of the relationship between torus-equivariant cohomology of Grassmannians and double Schur polynomials.