Deciding Positivity of Littlewood--Richardson Coefficients

Starting with Knutson and Tao's hive model [J. Amer. Math. Soc., 12 (1999), pp. 1055--1090] we characterize the Littlewood--Richardson coefficient cλ,μν of given partitions λ,μ,νⁿ as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding cλ,μν >0. This algorithm is easy to state and takes O(n³logν₁) arithmetic operations and comparisons. We further show that the capacity achieving hive flows can be seen as the vertices of a connected graph, which leads to new structural insights into Littlewood--Richardson coefficients.