# Schubert Derivations on the Infinite Wedge Power

## TL;DR

This paper extends Schubert derivations to the infinite exterior power of a free abelian group, linking them to vertex operators and the bosonic vertex representation of $gl_00$, revealing new algebraic structures in infinite-dimensional settings.

## Contribution

It introduces a novel extension of Schubert derivations to infinite wedge spaces and connects classical vertex operators with the cohomology of Grassmannians within this framework.

## Key findings

- Derived vertex operators from the integration by parts formula.
- Connected Schubert calculus with the bosonic vertex representation.
- Interpreted DJKM result as a limit case in infinite-dimensional cohomology.

## 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.}