  We relate the $m$-truncated Kronecker products of symmetric functions to the
semi-invariant rings of a family of quiver representations. We find cluster
algebra structures for these semi-invariant rings when $m=2$. Each {\sf
g}-vector cone ${\sf G}_{\Diamond_l}$ of these cluster algebras controls the
$2$-truncated Kronecker products for all symmetric functions of degree no
greater than $l$. As a consequence, each relevant Kronecker coefficient is the
difference of the number of the lattice points inside two rational polytopes.
We also give explicit description of all ${\sf G}_{\Diamond_l}$'s. As an
application, we compute some invariant rings.
