Asymptotic fluctuations of representations of the unitary groups

Details Asymptotic fluctuations of representations of the unitary groups Asymptotic fluctuations of representations of the unitary groups

2009-11-30

arxiv.org/abs/0911.5546v1

Mathematics

Representation Theory

Scientific paper

We study asymptotics of representations of the unitary groups U(n) in the limit n\to\infty and we show that in many aspects they behave like large random matrices. In particular, we show 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).

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