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Affine approach to quantum Schubert calculus
 Duke Math. J
"... This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur ..."
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This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3point GromovWitten invariants, which are the structure constants of the quantum cohomology ring. This construction implies three symmetries of the GromovWitten invariants of the Grassmannian with respect to the groups S3, (Z/nZ) 2, and Z/2Z. The last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q. Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of q with the highest power. We also discuss the affine nilTemperleyLieb
AFFINE SYMMETRIES OF THE EQUIVARIANT QUANTUM COHOMOLOGY RING OF RATIONAL HOMOGENEOUS SPACES
"... Abstract. Let X be a rational homogeneous space and let QH ∗ (X) × loc be the group of invertible elements in the small quantum cohomology ring of X localised in the quantum parameters. We generalise results of [2] and realise explicitly the map π1(Aut(X)) → QH ∗ (X) × loc described in [14]. We even ..."
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Abstract. Let X be a rational homogeneous space and let QH ∗ (X) × loc be the group of invertible elements in the small quantum cohomology ring of X localised in the quantum parameters. We generalise results of [2] and realise explicitly the map π1(Aut(X)) → QH ∗ (X) × loc described in [14]. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring QH ∗ T (X) × loc. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar [8]. It also uses Peterson’s unpublished result [11] — recently proved by Lam and Shimozono in [7] — on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring. 1.