In this note we observe that membership in moment cones of spaces of quiver representations can be decided in strongly polynomial time, for any acyclic quiver. This generalizes a recent result by Chindris-Collins-Kline for bipartite quivers. Their approach was to construct "multiplicity polytopes" with a geometric realization similar to the Knutson-Tao polytopes for tensor product multiplicities. Here we show that a less geometric but straightforward variant of their construction leads to such a multiplicity polytope for any acyclic quiver. Tardos' strongly polynomial time algorithm for combinatorial linear programming along with the saturation property then implies that moment cone membership can be decided in strongly polynomial time. The analogous question for semi-invariants remains open.

翻译：[摘要] 本文指出：对于任意无环箭图，其表示空间的矩锥成员关系可在强多项式时间内判定。该结果推广了Chindris-Collins-Kline近期关于二部箭图的研究。他们构造了具有几何实现的"重数多面体"，这类多面体与用于张量积重数的Knutson-Tao多面体具有相似性。本文证明：对任意无环箭图，采用其构造中较低几何化的直接变体即可得到相应的重数多面体。结合Tardos关于组合线性规划的强多项式时间算法与饱和性质，可导出矩锥成员关系在强多项式时间内可判定。关于半不变量的类似问题仍有待解决。