Deciding Positivity of Littlewood-Richardson Coefficients

TL;DR

This paper presents a polynomial-time algorithm to decide the positivity of Littlewood-Richardson coefficients using hive models, providing new structural insights and an efficient computational approach.

Contribution

It introduces a polynomial-time algorithm based on hive flows to determine positivity of Littlewood-Richardson coefficients and explores their structural properties.

Findings

Algorithm decides positivity in O(n^3 log ν_1) time

Hive flows correspond to vertices of a connected graph

Provides new structural insights into Littlewood-Richardson coefficients

Abstract

Starting with Knutson and Tao's hive model (in J. Amer. Math. Soc., 1999) we characterize the Littlewood-Richardson coefficient $c_{\lambda,\mu}^\nu$ of given partitions $\lambda,\mu,\nu\in N^n$ 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 $O(n^3 \log \nu_1)$ 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.