On Two Algebraic Realizations of Schubert Calculus

Schubert calculus on complex Grassmannians can be played by means of differential operators acting on Schur polynomials or Vertex Operators acting on exterior algebras. In this paper we develop this point of view systematically and complement it with a parallel exterior-algebra formalism, leading to what we call, respectively, the \emph{bosonic} and the \emph{fermionic} Schubert calculus. The two alluded realizations are related by (a finite type version of) the boson--fermion correspondence, thereby providing a unified framework connecting Schubert calculus and integrals on the Grassmannian, symmetric functions, exterior algebras and the representation theory of symmetric groups.