Minuscule Schubert calculus and the geometric Satake correspondence

We describe a relationship between work of Laksov, Gatto, and their collaborators on realizations of (generalized) Schubert calculus of Grassmannians, and the geometric Satake correspondence of Lusztig, Ginzburg, and Mirkovic and Vilonen. Along the way we obtain new proofs of equivariant Giambelli formulas for the ordinary and orthogonal Grassmannians, as well as a simple derivation of the rim-hook rule for computing in the equivariant quantum cohomology of the Grassmannian.