We introduce and study a new mathematical structure in the generalised
(quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert
calculus to a quantum integrable system known in the physics literature as the
asymmetric six-vertex model. Our approach offers a new perspective on already
established and well-studied special cases, for example equivariant K-theory,
and in addition allows us to formulate a conjecture on the so-far unknown case
of quantum equivariant K-theory.Comment: 57 pages, 10 figures; v2: some references added and some minor
changes; v3: abstract shortened, some typos corrected and a discussion of the
Bethe roots for the non-equivariant case added; v4: accepted versio
