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Cumulants of Jack symmetric functions and the $ b$-conjecture

Authors: Maciej Dołęga and Valentin Féray
Journal: Trans. Amer. Math. Soc. 369 (2017), 9015-9039
MSC (2010): Primary 05E05
Published electronically: September 7, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $ \psi (\boldsymbol {x}, \boldsymbol {y}, \boldsymbol {z}; t, 1+\beta )$ that might be interpreted as a continuous deformation of the generating series of rooted
hypermaps. They made the following conjecture: the coefficients of
$ \psi (\boldsymbol {x}, \boldsymbol {y}, \boldsymbol {z}; t, 1+\beta )$ in the power-sum basis are polynomials in $ \beta $ with non-negative integer coefficients (by construction, these coefficients are rational functions in $ \beta $).

We partially prove this conjecture, nowadays called the $ b$-conjecture, by showing that coefficients of $ \psi (\boldsymbol {x}, \boldsymbol {y}, \boldsymbol {z}; t, 1+ \beta )$ are polynomials in $ \beta $ with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter $ \alpha $ tends to 0, which may be of independent interest.

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  • [BH08] B. A. Bernevig and F. D. M. Haldane,
    Model fractional quantum Hall states and Jack polynomials,
    Phy. Rev. Lett. 100 (2008), no. 24, 246802.
  • [BJ07] D. R. L. Brown and D. M. Jackson, A rooted map invariant, non-orientability and Jack symmetric functions, J. Combin. Theory Ser. B 97 (2007), no. 3, 430-452. MR 2305897,
  • [DF16] Maciej Dołega and Valentin Féray, Gaussian fluctuations of Young diagrams and structure constants of Jack characters, Duke Math. J. 165 (2016), no. 7, 1193-1282. MR 3498866,
  • [DFŚ14] Maciej Dołega, Valentin Féray, and Piotr Śniady, Jack polynomials and orientability generating series of maps, Sém. Lothar. Combin. 70 (2013), Art. B70j, 50. MR 3378809
  • [Doł16a] Maciej Dołega, Strong factorization property of Macdonald polynomials and higher-order Macdonald's positivity conjecture, J. Algebraic Combin. 46 (2017), no. 1, 135-163. MR 3666415,
  • [Doł16b] Maciej Dołega, Top degree part in $ b$-conjecture for unicellular bipartite maps, Electron. J. Combin. 24 (2017), no. 3, Paper 3.224, 38.
  • [Fér13] Valentin Féray, Asymptotic behavior of some statistics in Ewens random permutations, Electron. J. Probab. 18 (2013), no. 76, 32. MR 3091722,
  • [FJMM02] 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. 23 (2002), 1223-1237. MR 1903954,
  • [GJ96] 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. MR 1325917,
  • [GR05] Adriano Garsia and Jeffrey B. Remmel, Breakthroughs in the theory of Macdonald polynomials, Proc. Natl. Acad. Sci. USA 102 (2005), no. 11, 3891-3894. MR 2139721,
  • [Jac71] Henry Jack, A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971), 1-18. MR 0289462
  • [JŁR00] Svante Janson, Tomasz Łuczak, and Andrzej Rucinski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000. MR 1782847
  • [Kad97] Kevin W. J. Kadell, The Selberg-Jack symmetric functions, Adv. Math. 130 (1997), no. 1, 33-102. MR 1467311,
  • [KS97] Friedrich Knop and Siddhartha Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9-22. MR 1437493,
  • [KV16] Andrei L. Kanunnikov and Ekaterina A. Vassilieva, On the matchings-Jack conjecture for Jack connection coefficients indexed by two single part partitions, Electron. J. Combin. 23 (2016), no. 1, Paper 1.53, 30. MR 3484758
  • [La09] Michael Andrew La Croix, The combinatorics of the Jack parameter and the genus series for topological maps, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)-University of Waterloo (Canada). MR 2714192
  • [Las08] Michel Lassalle, A positivity conjecture for Jack polynomials, Math. Res. Lett. 15 (2008), no. 4, 661-681. MR 2424904,
  • [Las09] Michel Lassalle, Jack polynomials and free cumulants, Adv. Math. 222 (2009), no. 6, 2227-2269. MR 2562783,
  • [LS59] V. P. Leonov and A. N. Sirjaev, On a method of semi-invariants, Theor. Probability Appl. 4 (1959), 319-329. MR 0123345
  • [LV95] Luc Lapointe and Luc Vinet, A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture, Internat. Math. Res. Notices 9 (1995), 419-424. MR 1360620,
  • [LV97] Luc Lapointe and Luc Vinet, Rodrigues formulas for the Macdonald polynomials, Adv. Math. 130 (1997), no. 2, 261-279. MR 1472319,
  • [LZ04] Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, with an appendix by Don B. Zagier, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004. MR 2036721
  • [Mac95] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., with contributions by A. Zelevinsky, Oxford Science Publications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. MR 1354144
  • [NŚ11] Jonathan Novak and Piotr Śniady, What is $ \dots$ a free cumulant?, Notices Amer. Math. Soc. 58 (2011), no. 2, 300-301. MR 2768121
  • [Sch47] Marcel-Paul Schutzenberger, Sur certains paramètres caractéristiques des systèmes d'événements compatibles et dépendants et leur application au calcul des cumulants de la répétition, C. R. Acad. Sci. Paris 225 (1947), 277-278. MR 0021668
  • [Śni15] P. Śniady,
    Top degree of Jack characters and enumeration of maps,
    preprint, 2015, arXiv:1506.06361.
  • [Sta89] Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76-115. MR 1014073,

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Additional Information

Maciej Dołęga
Affiliation: Wydział Matematyki i Informatyki, Uniwersytet im. Adama Mickiewicza, Collegium Mathematicum, Umultowska 87, 61-614 Poznań, Poland – and – Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Valentin Féray
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

Keywords: Jack symmetric functions, cumulants, Laplace-Beltrami operator
Received by editor(s): January 16, 2016
Received by editor(s) in revised form: January 23, 2017
Published electronically: September 7, 2017
Additional Notes: The first author was supported by Agence Nationale de la Recherche, grant ANR 12-JS02-001-01 “Cartaplus” and by NCN, grant UMO-2015/16/S/ST1/00420. The second author was partially supported by grant SNF-149461 “Dual combinatorics of Jack polynomials”.
Article copyright: © Copyright 2017 American Mathematical Society

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