QUANTUM K-THEORY OF GRASSMANNIANS

By Anders Skovsted Buch, Leonardo Constantin Mihalcea. 2008

84 citations recorded by the source corpus.

We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersec- tions 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 3 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 for- mula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6247eda1d297103cf84c94e0481dd855142145f2

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Finiteness of cominuscule quantum K-theory (2010, methodology): “Notice that the statement in [5] requires that the general fibers of f are rational.”
A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties (2016, methodology): “Equivariant quantum K-theory is the most general theory for which the associated Schubert structure constants have positivity properties that are either known [30, 51, 6, 3, 2] or conjectured [48, 49, 13].”