A Giambelli formula for even orthogonal Grassmannians

TL;DR

This paper establishes a Giambelli formula for even orthogonal Grassmannians, linking Schubert classes to special classes and revealing connections with odd cases and eta polynomials.

Contribution

It introduces a new Giambelli formula for even orthogonal Grassmannians and explores its relation to odd cases and eta polynomial algebra.

Findings

Derived a polynomial expression for Schubert classes in the cohomology ring.

Discovered a surprising relation between even and odd orthogonal Grassmannian calculus.

Connected eta polynomials with Schubert calculus on X.

Abstract

Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an even dimensional vector space equipped with a nondegenerate symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in the singular and quantum cohomology ring of X as a polynomial in certain special Schubert classes. Our analysis reveals a surprising relation between the Schubert calculus on even and odd orthogonal Grassmannians. We also study eta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X.