Quantum Pieri rules for isotropic Grassmannians

TL;DR

This paper develops combinatorial Pieri rules for the quantum cohomology of isotropic Grassmannians, enabling explicit calculations of Gromov-Witten invariants and ring presentations.

Contribution

It introduces new Pieri rules for both classical and quantum cohomology of isotropic Grassmannians, with explicit combinatorial formulas and ring presentations.

Findings

Established Pieri rules for classical cohomology.

Derived Pieri rules for small quantum cohomology.

Provided presentations of cohomology rings with generators and relations.

Abstract

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.