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 c(lambda)(nu),(mu) of given partitions lambda, mu, nu is an element of N-n as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding c(lambda)(nu),(mu) > 0. This algorithm is easy to state and takes O(n(3) log nu(1)) 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.