Large tensor products and Littlewood-Richardson coefficients

TL;DR

This paper provides a pure Lie theoretic proof for the density formula of the limiting measure associated with large tensor products and Littlewood-Richardson coefficients, connecting representation theory and measure theory.

Contribution

It offers a self-contained, Lie theoretic proof of the density formula for the limiting measure, expanding understanding of tensor product decompositions.

Findings

Derived the density formula for the limiting measure in tensor products

Connected Littlewood-Richardson coefficients with weight multiplicities

Provided a pure Lie theoretic approach to measure analysis

Abstract

The 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.