Computing fusion products of MV cycles using the Mirković–Vybornov isomorphism

Abstract

The fusion of two Mirković–Vilonen cycles is a degeneration of their product, defined using the Beilinson–Drinfeld Grassmannian. In this paper, we put in place a conceptually elementary approach to computing this product in type $A$. We do so by transferring the problem to a fusion of generalized orbital varieties using the Mirković–Vybornov isomorphism. As an application, we explicitly compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of ${\mathbf{GL}}_4$, confirming that all the cluster variables are contained in the Mirković–Vilonen basis.