  The affine Grassmannian of $SL_n$ admits an embedding into the Sato
Grassmannian, which further admits a Pl\"ucker embedding into the
projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman
describe the linear part of the ideal defining this embedding in terms of
certain elements of the dual of Fock space called "shuffles", and they
conjecture that these elements together with the Pl\"ucker relations suffice to
cut out the affine Grassmannian. We give a proof of this conjecture in two
steps: first we reinterpret the shuffles equations in terms of Frobenius twists
of symmetric functions. Using this, we reduce to a finite dimensional-problem,
which we solve. For the second step we introduce a finite-dimensional analogue
of the affine Grassmannians of $SL_n$, which we conjecture to be precisely the
reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces
invariant under a nilpotent operator.
