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Journal of Lie Theory 29 (2019), No. 4, 927--940
Copyright Heldermann Verlag 2019

Large Tensor Products and Littlewood-Richardson Coefficients

Evgeny Feigin
Department of Mathematics, HSE University, Moscow 119048, Russia
and: Skolkovo Institute of Science and Technology, Moscow 143026, Russia

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, self-contained 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.

Keywords: Littlewood-Richardson coefficients, distributions, limit shapes.

MSC: 17B10, 60B10.

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