  We study asymptotics of representations of the unitary groups U(n) in the
limit as n tends to infinity and we show that in many aspects they behave like
large random matrices. In particular, we prove that the highest weight of a
random irreducible component in the Kronecker tensor product of two irreducible
representations behaves asymptotically in the same way as the spectrum of the
sum of two large random matrices with prescribed eigenvalues. This agreement
happens not only on the level of the mean values (and thus can be described
within Voiculescu's free probability theory) but also on the level of
fluctuations (and thus can be described within the framework of higher order
free probability).
