Large Tensor Products and Littlewood-Richardson Coefficients

Littlewood-Richardson coefficients describe the decomposition of tensor products of irreducible representations of a simple Lie algebra into irreducibles.Assuming the number of factors is large, one gets a measure on the space of weights.This limiting measure was extensively studied by many authors.In particular, Kerov computed the corresponding density in a special case in type A and Kuperberg gave a formula for the general case.The goal of this paper is to give a short, selfcontained and pure Lie theoretic proof of the formula for the density of the limiting measure.Our approach is based on the link between the limiting measure induced by the Littlewood-Richardson coefficients and the measure defined by the weight multiplicities of the tensor products.