Representations of Unitary Groups and Free Convolution

Abstract

To each finite dimensional representation of a unitary group U(n) is associated a probability measure on the set of integers, depending on the highest weights which occur in this representation. We show that asymptotically for large n and large irreducible representations of U(n) the measure associated to the tensor product of two representations, or to the restriction of a representation to a subgroup U(m) with m comparable to n, can be expressed in terms of the measures associated to the first representations by means of the notion of free convolution (namely additive free convolution for the tensor product problem and multiplicative free convolution for the restriction problem).