Cluster Algebras and Semi-invariant Rings I. Triple Flags

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 ${\mathsf{G}}_n$ such that the generic cluster character maps its lattice points onto a basis of the 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.