A Giambelli formula for isotropic Grassmannians

TL;DR

This paper establishes a Giambelli formula for isotropic Grassmannians, expressing Schubert classes as polynomials in special classes, and connects theta polynomials with Schubert calculus and symmetric functions.

Contribution

It introduces a Giambelli formula for isotropic Grassmannians and links theta polynomials to Schubert calculus and symmetric functions.

Findings

Giambelli formula for isotropic Grassmannians derived

Theta polynomials shown to be special cases of Schubert polynomials

Expressed theta polynomials as positive combinations of Schur Q- and S-polynomials

Abstract

Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H^*(X,Z) as a polynomial in certain special Schubert classes. We study theta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X. Furthermore, we prove that theta polynomials are special cases of Billey-Haiman Schubert polynomials and use this connection to express the former as positive linear combinations of products of Schur Q-functions and S-polynomials.