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 $K^{\beta}_{\underline{\lambda}}$ be the multiplicity of the irreducible representation of $GL(\beta)$ of highest weight $\underline{\lambda}$ in the ring of polynomial functions on $rep(Q, \beta)$. We show that $K^{\beta}_{\underline{\lambda}}$ 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.

翻译：设$Q$为二分箭图，其顶点集为$Q_0$，且任意源顶点与任意汇顶点间的箭头数恒定。令$\beta=(\beta(x))_{x \in Q_0}$为$Q$的维数向量，其坐标均为正整数。设$rep(Q, \beta)$为$Q$的$\beta$维表示空间，$GL(\beta)$为作用于$rep(Q, \beta)$的基变换群，通过同时共轭作用。令$K^{\beta}_{\underline{\lambda}}$表示$GL(\beta)$的最高权$\underline{\lambda}$不可约表示在$rep(Q, \beta)$多项式函数环中的重数。我们证明$K^{\beta}_{\underline{\lambda}}$可表示为通过粘合两个Knutson-Tao蜂巢多面体所得多面体的整点数。进一步地，该多面体描述结合Derksen-Weyman关于箭图半不变量的饱和定理，使我们能够利用Tardos算法在强多项式时间内求解与$(Q,\beta)$相关的矩锥成员问题。