  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
$l,m\in \mathbb{N}$. 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 $g_{\mu,\nu}^\lambda$ with
$\lambda$ at most $m$ parts, we only need to count lattice points in at most
$m!$ fibre (rational) polytopes inside the ${\rm g}$-vector cone, which is
explicitly given.

Representation Theory;Commutative Algebra;Combinatorics;Group Theory