Explicit description of certain 3-point K-theoretic Gromov-Witten invariants for flag manifolds

Abstract

We give an explicit description, in terms of the quantum Bruhat graph, of the (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants $\langle \mathcal{O}(- \lambda), \mathcal{O}^{w}, \mathcal{O}_{u} \rangle_{d}$ for the (full) flag manifold $X = G/B$, where $\mathcal{O}(- \lambda)$ denotes the class in the (torus-equivariant) $K$-theory ring $K_{T}(X)$ of $X$ of the line bundle $\mathcal{O}_{X}(- \lambda) = G \times_{B} \mathbb{C}_{\lambda}$ over $X = G/B$ associated to a weight $\lambda \in W \varpi_i$ lying in the Weyl group orbit of a minuscule fundamental weight $\varpi_i$, and $\mathcal{O}_{u}$, $\mathcal{O}^{w}$ are the Schubert and opposite Schubert classes in $K_{T}(X)$ for $u, w \in W$. This result can be thought of as a partial generalization of the quantum $K$-theoretic divisor axiom, which we obtained in our previous work; our proof utilizes a generalization of the Chevalley formula in the (torus-equivariant) quantum $K$-theory ring $QK_{T}(X)$ of $X$, which computes the quantum product with the line bundle class $\mathcal{O}(- \lambda)$ associated to the weight $\lambda$ above.