Stein’s method and Plancherel measure of the symmetric group
HTML articles powered by AMS MathViewer
- by Jason Fulman PDF
- Trans. Amer. Math. Soc. 357 (2005), 555-570 Request permission
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.References
- David Aldous and Persi Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 413–432. MR 1694204, DOI 10.1090/S0273-0979-99-00796-X
- Richard Arratia, Larry Goldstein, and Louis Gordon, Poisson approximation and the Chen-Stein method, Statist. Sci. 5 (1990), no. 4, 403–434. With comments and a rejoinder by the authors. MR 1092983
- A. D. Barbour, Lars Holst, and Svante Janson, Poisson approximation, Oxford Studies in Probability, vol. 2, The Clarendon Press, Oxford University Press, New York, 1992. Oxford Science Publications. MR 1163825
- Philippe Biane, Representations of symmetric groups and free probability, Adv. Math. 138 (1998), no. 1, 126–181. MR 1644993, DOI 10.1006/aima.1998.1745
- Alexei Borodin, Andrei Okounkov, and Grigori Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), no. 3, 481–515. MR 1758751, DOI 10.1090/S0894-0347-00-00337-4
- Alexei Borodin and Grigori Olshanski, Harmonic functions on multiplicative graphs and interpolation polynomials, Electron. J. Combin. 7 (2000), Research Paper 28, 39. MR 1758654
- Percy Deift, Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), no. 6, 631–640. MR 1764262
- Persi Diaconis, James Allen Fill, and Jim Pitman, Analysis of top to random shuffles, Combin. Probab. Comput. 1 (1992), no. 2, 135–155. MR 1179244, DOI 10.1017/S0963548300000158
- Persi Diaconis and Laurent Saloff-Coste, Comparison techniques for random walk on finite groups, Ann. Probab. 21 (1993), no. 4, 2131–2156. MR 1245303
- Persi Diaconis and Mehrdad Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 2, 159–179. MR 626813, DOI 10.1007/BF00535487
- Alex Eskin and Andrei Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001), no. 1, 59–103. MR 1839286, DOI 10.1007/s002220100142
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- 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.
- Fulman, J., Card shuffling and the decomposition of tensor products, to appear in Pacific J. Math.
- Fulman, J., Stein’s method, Jack measure and the metropolis algorithm, preprint math. CO/0311290 at xxx.lanl.gov.
- Fulman, J., Stein’s method and nonreversible Markov chains (1997), to appear in 2000 Stein’s Method and Monte Carlo Markov Chains Conference Proceedings.
- Akihito Hora, Central limit theorem for the adjacency operators on the infinite symmetric group, Comm. Math. Phys. 195 (1998), no. 2, 405–416. MR 1637801, DOI 10.1007/s002200050395
- 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.
- Kurt Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), no. 1, 259–296. MR 1826414, DOI 10.2307/2661375
- Serguei Kerov, Gaussian limit for the Plancherel measure of the symmetric group, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 4, 303–308 (English, with English and French summaries). MR 1204294
- S. Kerov, The boundary of Young lattice and random Young tableaux, Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 133–158. MR 1363510, DOI 10.1007/bf02362775
- Andrei Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 20 (2000), 1043–1095. MR 1802530, DOI 10.1155/S1073792800000532
- Okounkov, A. and Pandharipande, R., Gromov-Witten theory, Hurwitz numbers, and matrix models, I, preprint math. AG/0101147 at xxx.lanl.gov.
- Yosef Rinott and Vladimir Rotar, On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics, Ann. Appl. Probab. 7 (1997), no. 4, 1080–1105. MR 1484798, DOI 10.1214/aoap/1043862425
- Bruce E. Sagan, The symmetric group, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Representations, combinatorial algorithms, and symmetric functions. MR 1093239
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
- J. Michael Steele, Probability theory and combinatorial optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. MR 1422018, DOI 10.1137/1.9781611970029
- Charles Stein, Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 7, Institute of Mathematical Statistics, Hayward, CA, 1986. MR 882007
- Charles Stein, A way of using auxiliary randomization, Probability theory (Singapore, 1989) de Gruyter, Berlin, 1992, pp. 159–180. MR 1188718
Additional Information
- Jason Fulman
- Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
- MR Author ID: 332245
- Email: fulman@math.pitt.edu
- Received by editor(s): May 28, 2003
- Received by editor(s) in revised form: July 7, 2003
- Published electronically: February 4, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 555-570
- MSC (2000): Primary 05E10; Secondary 60C05
- DOI: https://doi.org/10.1090/S0002-9947-04-03499-3
- MathSciNet review: 2095623