REPRESENTATIONS OF SYMMETRICAL GROUPS AND FREE PROBABILITY

We consider representations of symmetric groups S-q, for large ii. We give the asymptotic behaviour of the characters when the corresponding Young diagrams, rescaled by a factor q(-1/2), converge to some prescr ibed shape. Tills behaviour can be expressed in terms of the free cumu lants for a probability measure associated with the limit shape of the diagram. We also show that the basic operations of representation the ory, like taking tensor products, restriction, or induction, have a li miting behavior which can be described using the free probability theo ry of D. Voiculescu. (C) 1998 Academic Press.