  There are two intriguing statements regarding the quantum cohomology of
partial flag varieties. The first one relates quantum cohomology to the
affinisation of Lie algebras and the homology of the affine Grassmannian, the
second one connects it with the geometry of quiver varieties. The connection
with the affine Grassmannian was first discussed in unpublished work of
Peterson and subsequently proved by Lam and Shimozono. The second development
is based on recent works of Nekrasov, Shatashvili and of Maulik, Okounkov
relating the quantum cohomology of Nakajima varieties with integrable systems
and quantum groups. In this article we explore for the simplest case, the
Grassmannian, the relation between the two approaches. We extend the definition
of the integrable systems called vicious and osculating walkers to the
equivariant setting and show that these models have simple expressions in a
particular representation of the affine nil-Hecke ring. We compare this
representation with the one introduced by Kostant and Kumar and later used by
Peterson in his approach to Schubert calculus. We reveal an underlying quantum
group structure in terms of Yang-Baxter algebras and relate them to Schur-Weyl
duality. We also derive new combinatorial results for equivariant Gromov-Witten
invariants such as an explicit determinant formula.
