## Duke Mathematical Journal

### Quantum $K$-theory of Grassmannians

#### Abstract

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 $S_3$-symmetry. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.

#### Article information

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

Dates
First available in Project Euclid: 9 February 2011

https://projecteuclid.org/euclid.dmj/1297258908

Digital Object Identifier
doi:10.1215/00127094-2010-218

Mathematical Reviews number (MathSciNet)
MR2772069

Zentralblatt MATH identifier
1213.14103

#### Citation

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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