A Giambelli formula for isotropic Grassmannians

Anders S. Buch, Andrew Kresch, Harry Tamvakis

Published 2008-11-17, updated 2010-08-04Version 2

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.