Quasimaps, straightening laws, and quantum cohomology for the Lagrangian Grassmannian

TL;DR

This paper studies the algebraic and geometric properties of the Drinfel'd Lagrangian Grassmannian, establishing straightening laws for its Schubert subvarieties and deriving quantum cohomology intersection numbers.

Contribution

It introduces straightening laws for Schubert subvarieties of the Drinfel'd Lagrangian Grassmannian and connects these to quantum cohomology computations.

Findings

Schubert subvarieties are Cohen-Macaulay and Koszul

The ideal defining Schubert subvarieties is generated by straightening law polynomials

New derivation of intersection numbers in quantum cohomology

Abstract

The Drinfel'd Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the canonical embedding of the Lagrangian Grassmannian. We show that the defining ideal of any Schubert subvariety of the Drinfel'd Lagrangian Grassmannian is generated by polynomials which give a straightening law on an ordered set. Consequentially, any such subvariety is Cohen-Macaulay and Koszul. The Hilbert function is computed from the straightening law, leading to a new derivation of certain intersection numbers in the quantum cohomology ring of the Lagrangian Grassmannian.