Quantum Giambelli Formulas for Isotropic Grassmannians

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the authors’ cohomological Giambelli formulas. 0. Introduction Let E be an even (respectively, odd) dimensional complex vector space equipped with a nondegenerate skew-symmetric (respectively, symmetric) bilinear form. Let X denote the Grassmannian which parametrizes the isotropic subspaces of E of some fixed dimension. The cohomology ring H(X,Z) is generated by certain special Schubert classes, which for us are (up to a factor of two) the Chern classes of the universal quotient vector bundle over X. These special classes also generate the small quantum cohomology ring QH(X), a q-deformation of H(X,Z) whose structure constants are the three point, genus zero Gromov-Witten invariants of X. In [BKT3], we proved a Giambelli formula in H(X,Z), that is, a formula expressing a general Schubert class as an explicit polynomial in the special classes. Our goal in the present work is to extend this result to a formula that holds in QH(X). The quantum Giambelli formula for the usual type A Grassmannian was obtained by Bertram [Be], and is in fact identical to the classical Giambelli formula. In the case of maximal isotropic Grassmannians, the corresponding questions were answered in [KT1, KT2]. The main conclusions here are similar to those of loc. cit., provided that one uses the raising operator Giambelli formulas of [BKT3] as the classical starting point. For an odd orthogonal Grassmannian, we prove that the quantum Giambelli formula is the same as the classical one. The result is more interesting when X is the Grassmannian IG(n−k, 2n) parametrizing (n−k)dimensional isotropic subspaces of a symplectic vector space E of dimension 2n. Our theorem in this case states that the quantum Giambelli formula for IG(n−k, 2n) coincides with the classical Giambelli formula for IG(n + 1 − k, 2n + 2), provided that the special Schubert class σn+k+1 is replaced with q/2. Although the two theorems in this article are analogous to those of [KT1, KT2], their proofs are quite different. We prove the quantum Giambelli formula by using the quantum Pieri rule of [BKT2], in a manner similar to [Bu] and [BKT1, Remark Date: August 11, 2011. 2000 Mathematics Subject Classification. Primary 14N35; Secondary 05E15, 14M15, 14N15. The authors were supported in part by NSF Grants DMS-0603822 and DMS-0906148 (Buch), the Swiss National Science Foundation (Kresch), and NSF Grants DMS-0639033 and DMS0901341 (Tamvakis). 1 2 ANDERS SKOVSTED BUCH, ANDREW KRESCH, AND HARRY TAMVAKIS 3]. However, unlike the previously known examples, for non-maximal isotropic Grassmannians no explicit recursion formula for the cohomological Giambelli polynomials is available, other than that given by the Pieri rule itself. We circumvent this difficulty by showing that a suitable recursion exists (Proposition 3). We also make essential use of a ring homomorphism from the stable cohomology ring of X to QH(X) that is the identity on Schubert classes coming from H(X,Z). The existence of this map (Propositions 4 and 5) may be of independent interest. In a sequel to this paper, we will discuss the classical and quantum Giambelli formulas for even orthogonal Grassmannians. 1. Preliminary Results 1.1. Classical Giambelli for IG. Choose k ≥ 0 and consider the Grassmannian IG = IG(n − k, 2n) of isotropic (n − k)-dimensional subspaces of C, equipped with a symplectic form. A partition λ = (λ1 ≥ . . . ≥ λl) is k-strict if all of its parts greater than k are distinct integers. Following [BKT2], the Schubert classes on IG are parametrized by the k-strict partitions whose diagrams fit in an (n−k)×(n+k) rectangle, i.e. λ1 ≤ n+ k and l(λ) ≤ n− k; we denote the set of all such partitions by P(k, n). Given any partition λ ∈ P(k, n) and a complete flag of subspaces F • : 0 = F0 ( F1 ( · · · ( F2n = C 2n such that Fn+i = F ⊥ n−i for 0 ≤ i ≤ n, we have a Schubert variety Xλ(F•) := {Σ ∈ IG | dim(Σ ∩ Fpj(λ)) ≥ j ∀ 1 ≤ j ≤ l(λ)} , where l(λ) denotes the number of (non-zero) parts of λ and pj(λ) := n + k + j − λj − #{i < j : λi + λj > 2k + j − i}. This variety has codimension |λ| = ∑ λi and defines, via Poincaré duality, a Schubert class σλ = [Xλ(F•)] in H (IG,Z). The Schubert classes σλ for λ ∈ P(k, n) form a free Z-basis for the cohomology ring of IG. The special Schubert classes are defined by σr = [Xr(F•)] = cr(Q) for 1 ≤ r ≤ n+k, where Q denotes the universal quotient bundle over IG. The classical Giambelli formula for IG is expressed using Young’s raising operators [Y, p. 199]. We first agree that σ0 = 1 and σr = 0 for r < 0. For any integer sequence α = (α1, α2, . . .) with finite support and i < j, we set Rij(α) = (α1, . . . , αi + 1, . . . , αj − 1, . . .); a raising operator R is any monomial in these Rij ’s. Define mα = ∏ i σαi and Rmα = mRα for any raising operator R. For any k-strict partition λ, we consider the operator R = ∏ (1 −Rij) ∏ λi+λj>2k+j−i (1 + Rij) −1 where the first product is over all pairs i < j and second product is over pairs i < j such that λi +λj > 2k+ j− i. The main result of [BKT3] states that the Giambelli formula (1) σλ = R λ mλ holds in the cohomology ring of IG(n− k, 2n). QUANTUM GIAMBELLI FORMULAS FOR ISOTROPIC GRASSMANNIANS 3 1.2. Classical Pieri for IG. As is customary, we will represent a partition by its Young diagram of boxes; this is used to define the containment relation for partitions. Given two diagrams μ and ν with μ ⊂ ν, the skew diagram ν/μ (i.e., the set-theoretic difference ν r μ) is called a horizontal (resp. vertical) strip if it does not contain two boxes in the same column (resp. row). We say that the box [r, c] in row r and column c of a k-strict partition λ is k-related to the box [r, c] if |c − k − 1| + r = |c − k − 1| + r. For instance, the grey boxes in the following partition are k-related.