A Giambelli Formula for Even Orthogonal Grassmannians

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 classical 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. 0. Introduction Consider a complex vector space V of dimension N equipped with a nondegenerate symmetric form. Choose an integer m < N/2 and consider the Grassmannian OG = OG(m,N) parametrizing isotropic m-dimensional subspaces of V . Our aim in this paper is to prove a Giambelli formula that expresses the Schubert classes on OG as polynomials in certain special Schubert classes that generate the cohomology ring H(OG,Z). When N = 2n + 1 is odd, this was the main result of [BKT2]; what is new here concerns the even case N = 2n+ 2. The proof of our main theorem (Theorem 2) exploits the weight space decomposition of H(OG(m, 2n + 2),Q) induced by the natural involution of the Dynkin diagram of type Dn+1. We require the Giambelli formula for odd orthogonal Grassmannians from [BKT2] and a similar result for the (+1)-eigenspace of H(OG(m, 2n+ 2),Q), which is the subring generated by the Chern classes of the tautological vector bundles over OG. These ingredients combine to establish Theorem 2 thanks to a surprising new relation between the cohomology of even and odd orthogonal Grassmannians (Proposition 2). Define nonnegative integers K and k by the equations K = N − 2m = { 2k + 1 if N is odd, 2k if N is even. Observe that n+ k = N −m− 1. An integer partition λ = (λ1, . . . , λl) is k-strict if no part λi greater than k is repeated. Let λ be a k-strict partition whose Young diagram is contained in an m× (n+ k) rectangle. For 1 ≤ j ≤ m, let (1) pj(λ) = N −m+ j − λj −#{i ≤ j | λi + λj ≥ K + j − i and λi > k} , and notice that pj(λ) 6= n+ 1 for every j and λ. Date: March 29, 2012. 2000 Mathematics Subject Classification. Primary 14N15; Secondary 05E15, 14M15. The authors were supported in part by NSF Grant DMS-0906148 (Buch), the Swiss National Science Foundation (Kresch), and NSF Grant DMS-0901341 (Tamvakis). 1 2 ANDERS SKOVSTED BUCH, ANDREW KRESCH, AND HARRY TAMVAKIS An isotropic flag is a complete flag 0 = F0 ( F1 ( · · · ( FN = V of subspaces of V such that Fi = F ⊥ j whenever i+ j = N . For any fixed isotropic flag F• and any k-strict partition λ whose Young diagram is contained in an m× (n+ k) rectangle, we define a closed subset Yλ = Yλ(F•) ⊂ OG by setting (2) Yλ(F•) = {Σ ∈ OG | dim(Σ ∩ Fpj ) ≥ j for 1 ≤ j ≤ m} . If N is odd, the varieties Yλ are exactly the Schubert varieties in OG. If N is even, and k is not a part of λ, then Yλ is again a Schubert variety in OG. Otherwise, Yλ is a union of two Schubert varieties Xλ and X ′ λ, which will be defined below. The algebraic set Yλ has pure codimension |λ| = ∑ λi and determines a class [Yλ] in H(OG,Z). Consider the exact sequence of vector bundles over X = OG 0 → S → VX → Q → 0, where VX denotes the trivial bundle of rank N and S is the tautological subbundle of rank m. The Chern classes cp = cp(Q) of Q satisfy (3) cp = { [Yp] if p ≤ k, 2[Yp] if p > k. As in [BKT2], we will express our Giambelli formulas using Young’s raising operators [Y]. For any integer sequence α = (α1, α2, . . .) with finite support and i < j, we define Rij(α) = (α1, . . . , αi + 1, . . . , αj − 1, . . .). We also set cα = ∏ i cαi . If R is any finite monomial in the Rij ’s, then set Rcα = cRα; we stress that the operator R acts on the subscript α and not on the monomial cα itself. Given a k-strict partition λ we define the operator