Martingales and character ratios

Fulman, Jason

doi:10.1090/S0002-9947-06-03865-7

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Martingales and character ratios

Author: Jason Fulman
Journal: Trans. Amer. Math. Soc. 358 (2006), 4533-4552
MSC (2000): Primary 05E10; Secondary 60C05
Posted: April 11, 2006
MathSciNet review: 2231387
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Abstract: Some general connections between martingales and character ratios of finite groups are developed. As an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions. A generalization of these results is given for Jack measure on partitions. We also give a probabilistic proof of a result of Burnside and Brauer on the decomposition of tensor products.

[AlD] 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 (2000g:60013), http://dx.doi.org/10.1090/S0273-0979-99-00796-X
[Big] Norman Biggs, Algebraic graph theory, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. MR 1271140 (95h:05105)
[Bol] E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab. 10 (1982), no. 3, 672–688. MR 659537 (84a:60056)
[BOO] Alexei Borodin, Andrei Okounkov, and Grigori Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), no. 3, 481–515 (electronic). MR 1758751 (2001g:05103), http://dx.doi.org/10.1090/S0894-0347-00-00337-4
[BO] Alexei Borodin and Grigori Olshanski, 𝑍-measures on partitions and their scaling limits, European J. Combin. 26 (2005), no. 6, 795–834. MR 2143199 (2006d:60018), http://dx.doi.org/10.1016/j.ejc.2004.06.003
[De] Percy Deift, Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), no. 6, 631–640. MR 1764262 (2001g:05012)
[DSh] Persi Diaconis and Mehrdad Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 2, 159–179. MR 626813 (82h:60024), http://dx.doi.org/10.1007/BF00535487
[EO] 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 (2002g:32018), http://dx.doi.org/10.1007/s002220100142
[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.
[F1] Jason Fulman, Stein’s method and Plancherel measure of the symmetric group, Trans. Amer. Math. Soc. 357 (2005), no. 2, 555–570. MR 2095623 (2005e:05156), http://dx.doi.org/10.1090/S0002-9947-04-03499-3
[F2] Jason Fulman, Stein’s method, Jack measure, and the Metropolis algorithm, J. Combin. Theory Ser. A 108 (2004), no. 2, 275–296. MR 2098845 (2005h:60021), http://dx.doi.org/10.1016/j.jcta.2004.07.003
[F3] Jason Fulman, Card shuffling and the decomposition of tensor products, Pacific J. Math. 217 (2004), no. 2, 247–262. MR 2109933 (2005j:20008), http://dx.doi.org/10.2140/pjm.2004.217.247
[F4] Fulman, J., An inductive proof of the Berry-Esseen theorem for character ratios, to appear in Annals of Combin.
[F5] Fulman, J., Stein's method and random character ratios, preprint math.CO/0508291 at http://xxx.lanl.gov.
[GHJ] I. P. Goulden, J. L. Harer, and D. M. Jackson, A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves, Trans. Amer. Math. Soc. 353 (2001), no. 11, 4405–4427 (electronic). MR 1851176 (2002g:14035), http://dx.doi.org/10.1090/S0002-9947-01-02876-8
[Gr] Grams, W., Rates of convergence in the central limit theorem for dependent variables, Florida State University Thesis, 1972.
[Ha] Erich Haeusler, On the rate of convergence in the central limit theorem for martingales with discrete and continuous time, Ann. Probab. 16 (1988), no. 1, 275–299. MR 920271 (89a:60060)
[HH] P. Hall and C. C. Heyde, Martingale limit theory and its application, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. Probability and Mathematical Statistics. MR 624435 (83a:60001)
[Ho] 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 (99i:46058), http://dx.doi.org/10.1007/s002200050395
[Is] Isaacs, I.M., Character theory of finite groups, Dover Publications, Inc., New York, 1976.
[IO] Vladimir Ivanov and Grigori Olshanski, Kerov’s central limit theorem for the Plancherel measure on Young diagrams, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 93–151. MR 2059361 (2005d:05148), http://dx.doi.org/10.1007/978-94-010-0524-1_3
[J] Kurt Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), no. 1, 259–296. MR 1826414 (2002g:05188), http://dx.doi.org/10.2307/2661375
[K1] 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 (93k:20106)
[K2] 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 (96i:05177)
[K3] S. V. Kerov, Transition probabilities of continual Young diagrams and the Markov moment problem, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 32–49, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 27 (1993), no. 2, 104–117. MR 1251166 (95g:82045), http://dx.doi.org/10.1007/BF01085981
[K4] S. V. Kerov, Anisotropic Young diagrams and symmetric Jack functions, Funktsional. Anal. i Prilozhen. 34 (2000), no. 1, 51–64, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 34 (2000), no. 1, 41–51. MR 1756734 (2001f:05158), http://dx.doi.org/10.1007/BF02467066
[La] Michel Lassalle, Jack polynomials and some identities for partitions, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3455–3476 (electronic). MR 2055741 (2005i:05194), http://dx.doi.org/10.1090/S0002-9947-04-03500-7
[Ma] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144 (96h:05207)
[O1] Andrei Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 20 (2000), 1043–1095. MR 1802530 (2002c:15045), http://dx.doi.org/10.1155/S1073792800000532
[O2] Okounkov, A., The uses of random partitions, preprint math-ph/0309015 at http://xxx.lanl.gov.
[OP] Okounkov, A. and Pandaripandhe, R., Gromov-Witten theory, Hurwitz numbers, and Matrix models, I, preprint math.AG/0101147 at http://xxx.lanl.gov.
[Sa] Sagan, B., The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Springer-Verlag, New York, 1991.
[Se] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380 (56 #8675)
[ShSu] Shao, Q. and Su, Z., The Berry-Esseen bound for character ratios, preprint (2004).
[St] Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073 (90g:05020), http://dx.doi.org/10.1016/0001-8708(89)90015-7
[Su] Zhonggen Su, The law of the iterated logarithm for character ratios, Statist. Probab. Lett. 71 (2005), no. 4, 337–346. MR 2145501 (2006b:60054), http://dx.doi.org/10.1016/j.spl.2004.11.016

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