Duke Mathematical Journal

Quantum K-theory of Grassmannians

Anders S. Buch and Leonardo C. Mihalcea

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We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through three general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. We also compute the dual Schubert basis for this ring and show that its structure constants satisfy S3-symmetry. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.

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Duke Math. J., Volume 156, Number 3 (2011), 501-538.

First available in Project Euclid: 9 February 2011

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Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 19E08: $K$-theory of schemes [See also 14C35] 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14N15: Classical problems, Schubert calculus 14E08: Rationality questions [See also 14M20]


Buch, Anders S.; Mihalcea, Leonardo C. Quantum $K$ -theory of Grassmannians. Duke Math. J. 156 (2011), no. 3, 501--538. doi:10.1215/00127094-2010-218. https://projecteuclid.org/euclid.dmj/1297258908

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