Combinatorics in Quantum K-theory Schubert Calculus

I will discuss various applications of a combinatorial model for the (torus equivariant) quantum K-theory of flag manifolds G/B, called the quantum alcove model. This is a uniform model for all Lie types, based on Weyl group combinatorics. It first allows us to express a multiplication formula of Chevalley type in the quantum K-theory of G/B. This formula has several ramifications, including the solutions to some longstanding conjectures. The most recent such proof is of a conjecture due to Buch-Mihalcea, which can be viewed as a replacement of the "divisor axiom" for the quantum K-theory of G/B. It states that a K-theory Gromov-Witten invariant of Chevalley type (i.e., involving a divisor class) equals a classical (degree 0) invariant.

The talk includes joint work with Satoshi Naito, Daisuke Sagaki, and other collaborators.