Representations of Symmetric Groups and Free Probability

AbstractWe consider representations of symmetric groupsSqfor largeq. We give the asymptotic behaviour of the characters when the corresponding Young diagrams, rescaled by a factorq−1/2, converge to some prescribed shape. This behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram. We also show that the basic operations of representation theory, like taking tensor products, restriction, or induction, have a limiting behavior which can be described using the free probability theory of D. Voiculescu