  Let Gr be the affine Grassmannian for a connected complex reductive group G.
Let C_G be the complex vector space spanned by (equivalence classes of)
Mirkovic-Vilonen cycles in Gr. The Beilinson-Drinfeld Grassmannian can be used
to define a convolution product on MV-cycles, making C_G into a commutative
algebra. We show, in type A, that C_G isomorphic to C[N], the algebra of
functions on the unipotent radical N of a Borel subgroup of G; then each
MV-cycle defines a polynomial in C[N], which we call an MV-polynomial. We
conjecture that those MV-polynomials which are cluster monomials for a
Fomin-Zelevinsky cluster algebra structure on C[N] are naturally expressible as
determinants, and we conjecture a formula for many of them.
