The Algebra of Mirković-vilonen Cycles in Type a Jared Anderson and Mikhail Kogan

Let Gr be the affine Grassmannian for a connected complex reductive group G. Let CG be the complex vector space spanned by (equivalence classes of) Mirković-Vilonen cycles in Gr. The Beilinson-Drinfeld Grassmannian can be used to define a convolution product on MV-cycles, making CG into a commutative algebra. We show, in type A, that CG is 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 MVpolynomials 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. (Mathematics subject classification number: 14L35) This paper is dedicated to Robert MacPherson on the occasion of his 60th birthday.