International Congress of Mathematicians

772 P. Bianethis gives the order of magnitude of the character of a fixed conjugacy group for anA-balanced diagram.In [2] a proof of (4.1) has been given, using in an essential way the Jucys-Murphy operators. Another proof, leading to an exact formula for characters ofcycles due to S. Kerov [9], was shown to me later by A. Okounkov [15], see [5].5. Representations of large symmetric groupsThe asymptotic formula (4.1) shows in particular that irreducible characters ofsymmetric groups become asymptotically multiplicative i.e. for permutations withdisjoint supports 01 and 0 2 , one hasxAwz) = X^iJX^2) + Ofa- 1 -!!/ 2 ) (5.1)uniformly on A-balanced diagrams. Conversely, given a central, normalized, positivedefinite function on £„, a factorization property such as (5.1) implies that thepositive function is essentially an irreducible character [3]. Alore precisely, recallthat a central normalized positive definite function ip on £„ is a convex combinationof normalized characters, and as such it defines a probability measure on the setof Young diagrams. For any e, 5 > 0, for all n large enough, if an approximatefactorization such as (5.1) holds for ip, then there exists a curve OJ, such that themeasure on Young diagrams associated with ip puts a mass larger than 1 — 5 onYoung diagrams which lie in a neighbourhood of this curve, of width e\/n. Thereforeone can say that condition (5.1) on a positive definite function implies that therepresentation associated with this function is approximately isotypical, i.e. almostall Young diagrams occuring in the decomposition have a shape close to a certaindefinite curve.Using this fact it is possible to understand the asymptotic behaviour of severaloperations in representation theory. Consider for example the operation of induction.One starts with two irreducible representations of symmetric groups S ni , £„ 2 ,corresponding to two Young diagrams OJI and OJ 2 . One can then induce the productrepresentation u;i®u; 2 of S ni x £„ 2 to £„ 1+ „ 2 . This new representation is reducibleand the multiplicities of irreducible representations can be computed using a combinatorialdevice, the Littlewood-Richardson rule. This rule however gives littlelight on the asymptotic behaviour of the multiplicities. Using the factorizationconcentrationresult, one can prove that when rii and n 2 are very large, but ofthe same order of magnitude, then there exists a curve, which depends on ui andOJ 2 , and such that the typical Young diagram occuring in the decomposition of theinduced representation, is close to this curve. As we saw in section 4, one can associatea probability measure on the real line to any Young diagram. The descriptionof the typical shape of Young diagram which occurs in the decomposition of theinduced representation is easier if we use this correspondance between probabilitymeasuresand Young diagrams, indeed the probability measure associated with theshape of the typical Young diagram corresponds to the free convolution of the twoprobability measures [2].