Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones

Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta =(\beta (x){)}_{x\in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q,\beta )$ be the representation space of $\beta$-dimensional representations of $Q$ and $GL(\beta )$ the base change group acting on $rep(Q,\beta )$ be simultaneous conjugation. Let $K_{\underset{‾}{\lambda }}^{\beta }$ be the multiplicity of the irreducible representation of $GL(\beta )$ of highest weight $\underset{‾}{\lambda }$ in the ring of polynomial functions on $rep(Q,\beta )$. We show that $K_{\underset{‾}{\lambda }}^{\beta }$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman’s Saturation Theorem for quiver semi-invariants allows us to use Tardos’ algorithm to solve the membership problem for the moment cone associated to $(Q,\beta )$ in strongly polynomial time.