# On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

## TL;DR

This paper investigates the asymptotic behavior of irreducible representation multiplicities in large tensor products of simple Lie algebra representations, revealing universal patterns and dependencies on parameters.

## Contribution

It generalizes previous results by deriving the asymptotic distribution of irreducibles under Plancherel and character measures for large tensor products.

## Key findings

- Asymptotic distribution of irreducible components under Plancherel measure
- Universal asymptotic behavior of character measure near Plancherel measure
- Dependence of asymptotics on parameter degeneracy

## Abstract

In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotic of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.