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.