Schubert Calculus on a Grassmann Algebra

TL;DR

This paper presents a unified algebraic framework for the cohomology rings of Grassmannians, using derivations on exterior algebras, and connects it to universal splitting algebras of polynomials.

Contribution

It provides a unified presentation of classical, quantum, and equivariant cohomology rings of Grassmannians via generators and relations, linking to splitting algebra results.

Findings

Unified presentation of cohomology rings

Connection to universal splitting algebra

Derivations on exterior algebra characterize cohomology

Abstract

The ({\em classical}, {\em small quantum}, {\em equivariant}) cohomology ring of the grassmannian $G(k,n)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ ({\em Schubert Calculus on a Grassmann Algebra)}. Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. It also provides, by results of Laksov and Thorup, a presentation of the universal splitting algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$.