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Deciding Positivity of Littlewood-Richardson coefficients
Citation key BI-Deciding-Positivity-Of-Littlewood-Richardson-Coefficients
Author Peter Bürgisser and Christian Ikenmeyer
Pages 1639-1681
Year 2013
DOI 10.1137/120892532
Journal SIAM J. Discrete Math.
Volume 27
Number 4
Note Java-Applet: http://www3.math.tu-berlin.de/algebra/flowapplet/
Abstract Starting with Knutson and Tao's hive model (in J. Amer. Math. Soc., 1999) we characterize the Littlewood-Richardson coefficient $c_łambda,\mu^\nu$ of given partitions $łambda,\mu,\nu\in\mathbb N^n$ as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding $c_łambda,\mu^\nu >0$. This algorithm is easy to state and takes $O(n^3 łog \nu_1)$ 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.
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