Graded multiplicity in harmonic polynomials from the Vinberg setting

We consider Vinberg $\theta$-groups associated to a cyclic quiver on $r$ nodes. Let $K$ be the product of general linear groups associated to the nodes, acting naturally on $V=\oplus \text{Hom}(V_i,V_{i+1})$. We study the harmonic polynomials on $V$ in the specific case where $\dim V_i=2$ for all $i$. For each multigraded component of the harmonics, we give an explicit decomposition into irreducible representations of $K$, and additionally describe the multiplicities of each irreducible by counting integral points on certain faces of a polyhedron.