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The algebra of Mirkovic-Vilonen cycles in type A

Research paper by Jared E. Anderson, Mikhail Kogan

Indexed on: 06 May '05Published on: 06 May '05Published in: Mathematics - Algebraic Geometry



Abstract

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.