Duke Mathematical Journal

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Maciej Dołęga and Valentin Féray

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first- and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik, Kerov, Logan, and Shepp (for the first-order asymptotics) and Kerov (for the second-order asymptotics). This gives more evidence for the connection with the Gaussian β-ensemble, already suggested by a work of Matsumoto.

Our main tool is a polynomiality result for the structure constants of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interesting in itself and we give several other applications of it.

Article information

Source
Duke Math. J., Volume 165, Number 7 (2016), 1193-1282.

Dates
Received: 24 September 2014
Revised: 27 May 2015
First available in Project Euclid: 4 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1454594831

Digital Object Identifier
doi:10.1215/00127094-3449566

Mathematical Reviews number (MathSciNet)
MR3498866

Zentralblatt MATH identifier
1338.60017

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 05E05: Symmetric functions and generalizations

Keywords
random partitions Jack measure bulk fluctuations Jack polynomials polynomial functions on Young diagrams Stein’s method

Citation

Dołęga, Maciej; Féray, Valentin. Gaussian fluctuations of Young diagrams and structure constants of Jack characters. Duke Math. J. 165 (2016), no. 7, 1193--1282. doi:10.1215/00127094-3449566. https://projecteuclid.org/euclid.dmj/1454594831


Export citation

References

  • [1] K. Aker and M. B. Can, Generators of the Hecke algebra of $(S_{2n},B_{n})$, Adv. Math. 231 (2012), 2465–2483.
  • [2] G. W. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices, Cambridge Stud. Adv. Math. 118, Cambridge Univ. Press, Cambridge, 2010.
  • [3] J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119–1178.
  • [4] J. Baik, P. Deift, and E. Rains, A Fredholm determinant identity and the convergence of moments for random Young tableaux, Comm. Math. Phys. 223 (2001), 627–672.
  • [5] J. Baik and E. M. Rains, The asymptotics of monotone subsequences of involutions, Duke Math. J. 109 (2001), 205–281.
  • [6] Ph. Biane, Representations of symmetric groups and free probability, Adv. Math. 138 (1998), 126–181.
  • [7] Ph. Biane, Approximate factorization and concentration for characters of symmetric groups, Int. Math. Res. Not. IMRN 2001, no. 4 (2001), 179–192.
  • [8] Ph. Biane, “Characters of symmetric groups and free cumulants” in Asymptotic Combinatorics with Applications to Mathematical Physics (St. Petersburg, 2001), Lecture Notes in Math. 1815, Springer, Berlin, 2003, 185–200.
  • [9] A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481–515.
  • [10] A. Borodin and G. Olshanski, Z-measures on partitions and their scaling limits, European J. Combin. 26 (2005), 795–834.
  • [11] M. B. Can and Ş. Özden, Corrigendum to “Generators of the Hecke algebra of $(S_{2n},B_{n})$”, preprint, arXiv:1407.3700v1 [math.CO].
  • [12] A. Czyżewska-Jankowska, Topological expansion of the coefficients of zonal polynomials in genus one, preprint, arXiv:1108.3173v1 [math.CO].
  • [13] J. Dénes, The representation of a permutation as the product of a minimal number of transpositions and its connection with the theory of graphs, Publ. Math. Inst. Hung. Acad. Sci. 4 (1959), 63–70.
  • [14] M. Dołęga, V. Féray, and P. Śniady, Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations, Adv. Math. 225 (2010), 81–120.
  • [15] M. Dołęga and P. Śniady, Asymptotics of characters of symmetric groups: Structure of Kerov character polynomials, J. Combin. Theory Ser. A 119 (2012), 1174–1193.
  • [16] I. Dumitriu and A. Edelman, Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models, J. Math. Phys. 47 (2006), art. ID 063302, 36 pp.
  • [17] H. K. Farahat and G. Higman, The centres of symmetric group rings, Proc. Roy. Soc. London Ser. A 250 (1959), 212–221.
  • [18] B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, A differential ideal of symmetric polynomials spanned by Jack polynomials at $\beta=-(r-1)/(k+1)$, Int. Math. Res. Not. IMRN 2002, no. 23, 1223–1237.
  • [19] V. Féray, Combinatorial interpretation and positivity of Kerov’s character polynomials, J. Algebraic Combin. 29 (2009), 473–507.
  • [20] V. Féray, On complete functions in Jucys–Murphy elements, Ann. Comb. 16 (2012), 677–707.
  • [21] V. Féray and I. P. Goulden, A multivariate hook formula for labelled trees, J. Combin. Theory Ser. A 120 (2013), 944–959.
  • [22] V. Féray and P. Śniady, Zonal polynomials via Stanley’s coordinates and free cumulants, J. Algebra 334 (2011), 338–373.
  • [23] G. Frobenius, Über die Charaktere der symmetrischen Gruppe, Sitz. Konig. Preuss. Akad. Wissen 516 (1900), 148–166.
  • [24] J. Fulman, Stein’s method, Jack measure, and the Metropolis algorithm, J. Combin. Theory Ser. A 108 (2004), 275–296.
  • [25] J. Fulman, An inductive proof of the Berry–Esseen theorem for character ratios, Ann. Comb. 10 (2006), 319–332.
  • [26] J. Fulman, Stein’s method and random character ratios, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3687–3730.
  • [27] I. P. Goulden and D. M. Jackson, Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions, Trans. Amer. Math. Soc. 348 (1996), no. 3, 873–892.
  • [28] I. P. Goulden and D. M. Jackson, Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials, Canad. J. Math. 48 (1996), 569–584.
  • [29] I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, reprint of the 1983 original, Dover, Mineola, New York, 2004.
  • [30] I. P. Goulden and A. Rattan, An explicit form for Kerov’s character polynomials, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3669–3685.
  • [31] A. Goupil and G. Schaeffer, Factoring n-cycles and counting maps of given genus, European J. Combin. 19 (1998), 819–834.
  • [32] A. Hora, Central limit theorem for the adjacency operators on the infinite symmetric group, Comm. Math. Phys. 195 (1998), 405–416.
  • [33] V. N. Ivanov and S. V. Kerov, The algebra of conjugacy classes in symmetric groups, and partial permutations (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 256 (1999), 95–120; English translation in J. Math. Sci. (New York) 107 (2001), 4212–4230.
  • [34] V. Ivanov and G. Olshanski, “Kerov’s central limit theorem for the Plancherel measure on Young diagrams” in Symmetric Functions 2001: Surveys of Developments and Perspectives, NATO Sci. Ser. II Math. Phys. Chem. 74, Kluwer Acad., Dordrecht, 2002, 93–151.
  • [35] H. Jack, A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971), 1–18.
  • [36] K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), 151–204.
  • [37] K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), 259–296.
  • [38] K. Johansson, “Random permutations and the discrete Bessel kernel” in Random Matrix Models and their Applications, Math. Sci. Res. Inst. Publ. 40, Cambridge Univ. Press, Cambridge, 2001, 259–269.
  • [39] K. W. J. Kadell, The Selberg–Jack symmetric functions, Adv. Math. 130 (1997), 33–102.
  • [40] S. V. Kerov, The asymptotics of interlacing sequences and the growth of continual Young diagrams (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 205 (1993), 21–29, 179; English translation in J. Math. Sci. 80 (1996), no. 3, 1760–1767.
  • [41] S. Kerov, Gaussian limit for the Plancherel measure of the symmetric group, C. R. Math. Acad. Sci. Paris Sér. I 316 (1993), 303–308.
  • [42] S. Kerov, Transition probabilities of continual Young diagrams and the Markov moment problem (in Russian), Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 32–49, 96; English translation in Funct. Anal. Appl. 27 (1993), no. 2, 104–117.
  • [43] S. Kerov, Anisotropic Young diagrams and Jack symmetric functions (in Russian), Funktsional. Anal. i Prilozhen. 34 (2000), no. 1, 51–64, 96; English translation in Funct. Anal. Appl. 34 (2000), 45–51.
  • [44] S. Kerov, talk at the Institute Henri Poincaré, Paris, January, 2000.
  • [45] S. V. Kerov and G. Olshanski, Polynomial functions on the set of Young diagrams, C. R. Math. Acad. Sci. Paris Sér. I 319 (1994), 121–126.
  • [46] F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeros, Int. Math. Res. Not. IMRN 1996, no. 10, 473–486.
  • [47] F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 9–22.
  • [48] S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and their Applications, with appendix “Low-Dimensional Topology, II” by D. B. Zagier, Encyclopaedia Math. Sci. 141, Springer, Berlin, 2004.
  • [49] L. Lapointe and L. Vinet, A Rodrigues formula for the Jack polynomials and the Macdonald–Stanley conjecture, Int. Math. Res. Not. IMRN 1995, no. 9, 419–424.
  • [50] M. Lassalle, A positivity conjecture for Jack polynomials, Math. Res. Lett. 15 (2008), 661–681.
  • [51] M. Lassalle, Jack polynomials and free cumulants, Adv. Math. 222 (2009), 2227–2269.
  • [52] B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Adv. Math. 26 (1977), 206–222.
  • [53] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
  • [54] S. Matsumoto, Jack deformations of Plancherel measures and traceless Gaussian random matrices, Electron. J. Combin. 15 (2008), research paper 149, 18 pp.
  • [55] S. Matsumoto, Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures, Ramanujan J. 26 (2011), 69–107.
  • [56] M. L. Mehta, Random Matrices, 3rd ed., Pure Appl. Math. (Amsterdam) 142, Elsevier/Academic Press, Amsterdam, 2004.
  • [57] A. Okounkov, Random matrices and random permutations, Int. Math. Res. Not. IMRN 2000, no. 20, 1043–1095.
  • [58] A. Okounkov, “The uses of random partitions” in Fourteenth International Congress on Mathematical Physics, World Sci., Hackensack, N.J., 2003, 379–403.
  • [59] A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications, Math. Res. Lett. 4 (1997), 69–78.
  • [60] G. Olshanski, Plancherel averages: Remarks on a paper by Stanley, Electron. J. Combin. 17 (2010), no. 1, research paper 43, 16 pp.
  • [61] P. Petrullo and D. Senato, Explicit formulae for Kerov polynomials, J. Algebraic Combin. 33 (2011), 141–151.
  • [62] J. A. Ramírez, B. Rider, and B. Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc. 24 (2011), 919–944.
  • [63] G. Reinert and A. Röllin, Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition, Ann. Probab. 37 (2009), 2150–2173.
  • [64] B. E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd ed., Grad. Texts in Math. 203, Springer, New York, 2001.
  • [65] Q.-M. Shao and Z.-G. Su, The Berry-Esseen bound for character ratios, Proc. Amer. Math. Soc. 134 (2006), 2153–2159.
  • [66] P. Śniady, Asymptotics of characters of symmetric groups, genus expansion and free probability, Discrete Math. 306 (2006), 624–665.
  • [67] P. Śniady, personal communication, May 2014.
  • [68] R. Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994), 611–628.
  • [69] R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76–115.
  • [70] C. Stein, Approximate Computation of Expectations, Lect. Notes Monogr. Ser 7, Inst. Math. Stat., Hayward, Calif., 1986.
  • [71] O. Tout, Structure coefficients of the Hecke algebra of $(S_{2n},B_{n})$, Electron. J. Combin. 21 (2014), no. 4, paper 4.35, 41 pp.
  • [72] E. A. Vassilieva, Polynomial properties of Jack connection coefficients and generalization of a result by Dénes, J. Algebraic Combin. 42 (2015), 51–71.
  • [73] A. M. Vershik and S. V. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables (in Russian), Dokl. Acad. Nauk. SSSR 233 no. 6 (1977), 1024–1027; English translation in Soviet Math. Dokl. 233 no. 1–6 (1977), 527–531.
  • [74] D. Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), 323–346.