  We study the asymptotics of representations of a fixed compact Lie group. We
prove that the limit behavior of a sequence of such representations can be
described in terms of certain random matrices; in particular operations on
representations (for example: tensor product, restriction to a subgroup)
correspond to some natural operations on random matrices (respectively: sum of
independent random matrices, taking the corners of a random matrix). Our method
of proof is to treat the canonical block matrix associated to a representation
as a random matrix with non-commutative entries.
