Cluster algebras and semi-invariant rings I. Triple flags

Abstract

We prove that each semi-invariant ring of the complete triple flag of length n is an upper cluster algebra associated to an ice hive quiver. We find a rational polyhedral cone Gn such that the generic cluster character maps its lattice points onto a basis of the upper cluster algebra. As an application, we use the cluster algebra structure to find a special minimal set of generators for these semi-invariant rings when n is small.