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

TL;DR

This paper introduces an elementary method to compute fusion products of MV cycles in type A by leveraging the Mirkovic-Vybornov isomorphism, with applications to cluster algebra structures in GL4.

Contribution

It provides a new, simplified approach to calculating MV cycle fusions using the Mirkovic-Vybornov isomorphism, connecting geometric and combinatorial aspects.

Findings

Explicit computation of all cluster exchange relations in GL4

Confirmation that all cluster variables lie in the MV basis

Demonstration of the elementary approach's effectiveness

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 $GL_4$, confirming that all the cluster variables are contained in the Mirkovic-Vilonen basis.