An identity involving symmetric polynomials and the geometry of Lagrangian Grassmannians

TL;DR

This paper establishes a symmetric polynomial identity that unlocks new geometric and enumerative insights into Lagrangian Grassmannians, including formulas for characteristic classes, degrees, and Gromov--Witten invariants.

Contribution

It introduces a novel symmetric polynomial identity and applies it to derive formulas and relations for the geometry and enumerative invariants of Lagrangian Grassmannians.

Findings

Derived a formula for characteristic class integrals over Lagrangian Grassmannians.

Established a relation between Lagrangian and ordinary Grassmannians.

Computed Gromov--Witten invariants and Schubert structure constants explicitly.

Abstract

We first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of a characteristic class of the tautological sub-bundle. Moreover, a relation to that over the ordinary Grassmannian and its application to the degree formula for the Lagrangian Grassmannian are given. Finally, we present further applications to the computation of Schubert structure constants and three-point, degree $1$, genus $0$ Gromov--Witten invariants of the Lagrangian Grassmannian. Some examples together with explicit computations are presented.