Schubert Polynomials for the Affine Grassmannian of the Symplectic Group
نویسندگان
چکیده
We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur’s P and Q functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.
منابع مشابه
Schubert Polynomials for the Affine Grassmannian
Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the k-Schur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on non-commutative k-Schur functions.
متن کاملA 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 combination...
متن کاملQuantum Cohomology of the Lagrangian Grassmannian
Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V . We give a presentation for the (small) quantum cohomology ring QH∗(LG) and show that its multiplicative structure is determined by the ring of Q̃-polynomials. We formulate a ‘quantum Schubert calculus’ which includes quantum Pieri and Giambelli formulas, as well as algor...
متن کاملQuantum Cohomology of G/p and Homology of Affine Grassmannian
Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH∗(G/P ) of a flag variety is, up to localization, a quotient of the homology H∗(GrG) of the affine Grassmannian GrG of G. As a consequence, all three-point genus zero Gromov-Witten invariants of G/P are identified wi...
متن کاملK-theory Schubert calculus of the affine Grassmannian
We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G= SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of sym...
متن کامل