A Giambelli Formula for Isotropic Grassmannians

LetX 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 introduce and study theta polynomials, a family of polynomials which are positive linear combinations of products of Schur Qand S-functions, and whose algebra agrees with Schubert calculus on X. 0. Introduction Let G = G(m,N) denote the Grassmannian of m-dimensional subspaces of complex affine N -space. To each integer partition λ = (λ1, . . . , λm) whose Young diagram is contained in an m × (N −m) rectangle, we associate a Schubert class σλ in the cohomology ring of G. The special Schubert classes σ1, . . . , σN−m are the Chern classes of the universal quotient bundle Q over G(m,N); they generate the ring H(G,Z). The classical Giambelli formula [G] (1) σλ = det(σλi+j−i)i,j is an explicit expression for σλ as a polynomial in the special classes; as is customary, we agree here and in later formulas that σ0 = 1 and σr = 0 for r < 0. The relation between the Schubert calculus on the Grassmannian G(m,N) and the algebra of Schur’s S-functions sλ (originally defined by Cauchy [C] and Jacobi [J]) is well known. Given x = (x1, x2, . . .) a countably infinite set of commuting independent variables, we define the elementary symmetric functions er(x) by the formal relation ∞ ∏