Schubert Derivations on the Infinite Wedge Power

Abstract

The {\em Schubert derivation} is a distinguished Hasse-Schmidt derivation on the exterior algebra of a free abelian group, encoding the formalism of Schubert calculus for all Grassmannians at once. The purpose of this paper is to extend the Schubert derivation to the infinite exterior power of a free ${\mathbb Z}$-module of infinite rank (fermionic Fock space). Classical vertex operators naturally arise from the {\em integration by parts formula}, that also recovers the generating function occurring in the {\em bosonic vertex representation} of the Lie algebra $gl_\infty({\mathbb Z})$, due to Date, Jimbo, Kashiwara and Miwa (DJKM). In the present framework, the DJKM result will be interpreted as a limit case of the following general observation: the singular cohomology of the complex Grassmannian $G(r,n)$ is an irreducible representation of the Lie algebra of $n\times n$ square matrices.}