Transactions of the American Mathematical Society

Author(s): Jason Fulman
Journal: Trans. Amer. Math. Soc. 357 (2005), 555-570.
MSC (2000): Primary 05E10; Secondary 60C05
Posted: February 4, 2004
Retrieve article in: PDF DVI PostScript

Abstract: We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.

[AlD]: Aldous, D. and Diaconis, P., Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. AMS (N.S.) 36 (1999), 413-432. MR 2000g:60013
[ArGG]: Arratia, R., Goldstein, L. and Gordon, L., Poisson approximation and the Chen-Stein method. Statist. Sci. 5 (1990), 403-434. MR 92e:62036
[BHJ]: Barbour, A.D., Holst, L., and Janson, S., Poisson approximation. Oxford Science Publications. Clarendon Press, Oxford University Press, New York, 1992. MR 93g:60043
[Bi]: Biane, P., Representations of symmetric groups and free probability, Adv. Math. 138 (1998), 126-181. MR 2001b:05225
[BOO]: Borodin, A., Okounkov, A., and Olshanski, G., Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481-515. MR 2001g:05103
[BO]: Borodin, A. and Olshanski, G., Harmonic functions on multiplicative graphs and interpolation polynomials, Electron. J. Combin. 7 (2000), Research paper 28, 39 pages (electronic). MR 2001f:05160
[De]: Deift, P., Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), 631-640. MR 2001g:05012
[DFP]: Diaconis, P., Fill, J., and Pitman, J., Analysis of top to random shuffles, Combin. Probab. Comput. 1 (1992), 135-155. MR 93f:60011
[DSa]: Diaconis, P. and Saloff-Coste, L., Comparison techniques for random walk on finite groups, Ann. Probab. 21 (1993), 2131-2156. MR 95a:60009
[DSh]: Diaconis, P. and Shahshahani, M., Generating a random permutation with random transpositions, Z. Wahr. Verw. Gebiete 57 (1981), 159-179. MR 82h:60024
[EO]: Eskin, A. and Okounkov, A., Asymptotics of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001), 59-103. MR 2002g:32018
[Fe]: Feller, W., An introduction to probability theory and its applications, 2nd edition, John Wiley and Sons, Inc., Canada, 1957. MR 19:466a
[Fr]: Frobenuis, F., Uber die charaktere der symmetrischen gruppe, Sitz. Konig. Preuss. Akad. Wissen. (1900), 516-534; Gesammelte abhandlungen III, Springer-Verlag, Heidelberg, 1968, 148-166.
[Fu1]: Fulman, J., Card shuffling and the decomposition of tensor products, to appear in Pacific J. Math.
[Fu2]: Fulman, J., Stein's method, Jack measure and the metropolis algorithm, preprint math. CO/0311290 at xxx.lanl.gov.
[Fu3]: Fulman, J., Stein's method and nonreversible Markov chains (1997), to appear in 2000 Stein's Method and Monte Carlo Markov Chains Conference Proceedings.
[H]: Hora, A., Central limit theorem for the adjacency operators on the infinite symmetric group, Comm. Math. Phys. 195 (1998), 405-416. MR 99i:46058
[IO]: Ivanov, V. and Olshanski, G., Kerov's central limit theorem for the Plancherel measure on Young diagrams, in Symmetric Functions 2001: Surveys of developments and perspectives, Kluwer Academic Publishers, Dodrecht, 2002.
[J]: Johansson, K., Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), 259-296. MR 2002g:05188
[K1]: Kerov. S.V., Gaussian limit for the Plancherel measure of the symmetric group, Compt. Rend. Acad. Sci. Paris, Serie I, 316 (1993), 303-308. MR 93k:20106
[K2]: Kerov, S.V., The boundary of Young lattice and random Young tableaux, Formal power series and algebraic combinatorics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 24, Amer. Math. Soc., Providence, RI, (1996), 133-158. MR 96i:05177
[O]: Okounkov, A., Random matrices and random permutations, Internat. Math. Res. Notices 20 (2000), 1043-1095. MR 2002c:15045
[OP]: Okounkov, A. and Pandharipande, R., Gromov-Witten theory, Hurwitz numbers, and matrix models, I, preprint math. AG/0101147 at xxx.lanl.gov.
[RR]: Rinott, Y. and Rotar, V., On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics, Ann. of Appl. Probab. 7 (1997), 1080-1105. MR 99g:60050
[Sa]: Sagan, B., The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Springer-Verlag, New York, 1991. MR 93f:05102
[Sta]: Stanley, R., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. MR 2000k:05026
[Ste]: Steele, J. M., Probability theory and combinatorial optimization, Society for Industrial and Applied Mathematics, Philadelphia, 1997. MR 99d:60002
[Stn1]: Stein, C., Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes, Volume 7, 1986. MR 88j:60055
[Stn2]: Stein, C., A way of using auxiliary randomization. Probability theory (Singapore, 1989), (1992), 159-180. MR 93i:60004

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05E10, 60C05

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS