Computing fusion products of MV cycles using the Mirkovic-Vybornov isomorphism

Abstract

The fusion of two Mirkovic-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 Mirkovic-Vybornov isomorphism. As an application, we explicitly compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of $G{L}_4$, confirming that all the cluster variables are contained in the Mirkovic-Vilonen basis.

33 pages

References in corpus (3)

• Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
• Knuth relations, tableaux and MV-cycles
• Comparing two perfect bases