International Congress of Mathematicians

Free Probability and Combinatorics 773There are analogous results for the restriction of representations from largesymmetric groups to smaller ones. There the corresponding operation on probabilitymeasureis called the free compression, it corresponds at the level of the large matrixapproximation, to taking a random matrix with prescribed eigenvalue distribution,as in section 2, and extracting a square submatrix. Finally there are also results forKronecker tensor products of representations. Here a central role is played by thewell known Kerov-Vershik limit shape, whose associated probability measure is thesemi-circle distribution with density 7y-\j4 — x 2 on the interval [—2,2], see [2].References[I] P. Biane, Some properties of crossings and partitions. Discrete Math. 175(1997), no. 1-3, 41-53.[2] P. Biane, Representations of symmetric groups and free probability. Adv. Math.138 (1998), no. 1, 126-181.[3] P. Biane, Approximate factorization and concentration for characters of symmetricgroups. Internat. Math. Res. Notices no. 4 (2001), 179-192.[4] P. Biane, Entropie libre et algèbres d'opérateurs. Séminaire Bourbaki Exposé889, Juin 2001.[5] P. Biane, Free cumulants and characters of symmetric groups. Preprint, 2001.[6] B. Collins, Aloments and cumulants of polynomial random variables on unitarygroupsPreprint, 2002.[7] L. Al. Ge, these Proceedings.[8] S. V. Kerov, Transition probabilities of continual Young diagrams and theMarkov moment problem Funct. Anal. Appi. 27 (1993), 104-117.[9] S.Kerov, talk at IHP, January 2000.[10] G. Kreweras, Sur les partitions non croises d'un cycle. Discrete Math. 1 (1972),no. 4, 333-350.[II] A. Nica, R. Speicher, Commutators of free random variables, Duke Math. J.92 (1998), no. 3, 553-592.[12] A. Nica, R. Speicher, On the multiplication of free A r -tuples of noneommutativerandom variables, Amer. J. Math. 118 (1996), no. 4, 799-837.[13] A. Nica, R Speicher, iï-diagonal pairs—a common approach to Haar unitariesand circular elements. Free probability theory (Waterloo, ON, 1995), 149-188,Fields Inst. Commun., 12, Amer. Alath. Soc, Providence, RI, 1997.[14] A. Nica, D. Shlyakhtenko, R. Speicher, iï-diagonal elements and freeness withamalgamation. Canad. J. Math. 53 (2001), no. 2, 355-381.[15] A. Okounkov, private communication.[16] G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Albiusfunctions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 340-368.[17] A. N. Shiryaev, Probability Graduate texts in Alathematics 95, Springer, 1991.[18] P. Sniady, R. Speicher, Continuous family of invariant subspaces for i?-diagonaloperators. Invent. Math., 146 (2001), no. 2, 329-363.[19] R. Speicher, Alultiplicative functions on the lattice of noncrossing partitionsand free convolution. Math. Ann., 298 (1994), no. 4, 611-628.