Deciding Positivity of Littlewood--Richardson Coefficients

Abstract

Starting with Knutson and Tao's hive model [J. Amer. Math. Soc., 12 (1999), pp. 1055--1090] we characterize the Littlewood--Richardson coefficient ${c_{\lambda,\mu}^{\nu}}$ of given partitions $\lambda,\mu,\nu\in\mathbb{N}^n$ as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding ${c_{\lambda,\mu}^{\nu}} >0$. This algorithm is easy to state and takes $\mathcal{O}(n^3\log\nu_1)$ 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.