Quantum $K$-theoretic divisor axiom for flag manifolds

Abstract: We prove an identity for (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants of full or partial flag manifolds, which can be thought of as a replacement for the ``divisor axiom'' for the (torus-equivariant) quantum $K$-theory of flag manifolds. This identity enables us to compute the (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants defined by two Schubert classes and a divisor Schubert class in the (torus-equivariant) ordinary $K$-theory ring of flag manifolds. We prove this identity by making use of the Chevalley formula for the (torus-equivariant) quantum $K$-theory ring of flag manifolds, which is described in terms of the quantum Bruhat graph.