A Giambelli formula for isotropic Grassmannians

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.

39 pages; improvements and corrections made to the exposition

References in corpus (3)

• Quantum Pieri rules for isotropic Grassmannians
• Quantum Giambelli formulas for isotropic Grassmannians
• A Giambelli formula for classical $G/P$ spaces

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• Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D
• Double theta polynomials and equivariant Giambelli formulas
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• Quantum Giambelli formulas for isotropic Grassmannians
• $ı$Hall algebra of Jordan quiver and $ı$Hall-Littlewood functions
• Quantum Pieri rules for tautological subbundles
• A tableau formula for eta polynomials
• A Giambelli formula for classical $G/P$ spaces
• Degeneracy locus formulas for amenable Weyl group elements