Cluster algebras, invariant theory, and Kronecker coefficients II

Abstract

We prove that the semi-invariant ring of the standard representation space of the l-flagged m-arrow Kronecker quiver is an upper cluster algebra for any l,m∈N. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient gμ,νλ with λ at most m parts, we only need to count lattice points in at most m! fiber (rational) polytopes inside the g-vector cone, which is explicitly given.