Abstract
In this paper, we are interested in the asymptotic size of the rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order n, so it does not fit in the context of the work of Biane (Int Math Res Notices 4:179–192, 2001) and Śniady (Probab. Theory Relat Fields 136:263–297, 2006). Using the theory of polynomial functions on Young diagrams of Kerov and Olshanski, we are able to compute explicitly the first- and second-order asymptotics of the length of the first rows. Our method also works for other measures, for example those coming from Schur–Weyl representations.
References
Baik J., Deift P., Johansson K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999)
Baik J., Deift P., Johansson K.: On the distribution of the length of the second row of a Young diagram under Plancherel measure. Geom. Funct. Anal. 10(4), 702–731 (2000)
Biane P.: Approximate factorization and concentration for characters of symmetric groups. Int. Math. Res. Notices 4, 179–192 (2001)
Billingsley P.: Convergence of Probability Measures. Wiley, London (1969)
Borodin A.M.: The law of large numbers and the central limit theorem for the Jordan normal form of large triangular matrices over a finite field. J. Math. Sci. 96(5), 3455–3471 (1999)
Borodin A., Okounkov A., Olshanski G.: Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc. 13(3), 481–515 (2000)
Brillinger D.: The calculation of cumulants via conditioning. Ann. Inst. Stat. Math. 21, 375–390 (1969)
Collins, B., Śniady, P.: Asymptotic fluctuations of representations of the unitary groups. arXiv:0911.5546 (2009)
Geck M.: A note on harish-chandra induction. Manuscr. Math. 80, 393–401 (1993)
Geck M., Pfeiffer G.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, volume 21 of London Mathematical Society Monographs. Oxford University Press, New York (2000)
Howlett R.B., Lehrer G.I.: Induced cuspidal representations and generalised Hecke rings. Invent. Math. 58, 37–64 (1980)
Ivanov, V., Olshanski, G.: Kerov’s central limit theorem for the Plancherel measure on Young diagrams. In: Symmetric Functions 2001: Surveys of Developments and Perspectives, vol.74 of NATO Science Series II. Mathematics, Physics and Chemistry, pp. 93–151 (2002)
Iwahori N.: On the structure of the Hecke ring of a Chevalley group over a finite field. J Fac. Sci. Tokyo Univ. 10, 215–236 (1964)
Johansson K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153(2), 259–296 (2001)
Kerov S.V.: q-analogue of the hook walk algorithm and random Young tableaux. Funct. Anal. Appl. 26(3), 179–187 (1992)
Kerov, S.V.: A differential model for the growth of Young diagrams. In: Proceedings of the St. Petersburg Mathematical Society, 4, 165–192, 1996. (English translation: American Mathematical Society Translations, Series 2, 188, 1998)
Kerov S.V.: Asymptotic representation theory of the symmetric group and its applications in analysis, volume 219 of Trans. Math. Monographs. AMS Press, New York (2003)
Kerov S.V., Okounkov A., Olshanski G.: The boundary of the Young graph with Jack edge multiplicities. Int. Math. Res. Notices. 1998(4), 173 (1998)
Kerov S.V., Olshanski G.: Polynomial functions on the set of Young diagrams. Comptes Rendus Acad. Sci. Paris Série I. 319, 121–126 (1994)
Kerov S.V., Olshanski G., Vershik A.M.: Harmonic analysis on the infinite symmetric group. Invent. Math. 158, 551–642 (2004)
Kerov, S.V., Vershik, A.M.: Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux. Doklady AN SSSR, 233(6):1024–1027, 1977. (English translation: Soviet Mathematics Doklady 18, 527–531 1977)
Kerov, S.V., Vershik, A.M.: Asymptotic theory of characters of the symmetric group. Funkts. Anal. i Prilozhen, 15, 15–27, 1981. English translation: Funct. Anal. Appl. 15, 246–255 (1982)
Leonov V.P., Sirjaev A.N.: On a method of semi-invariants. Theor. Prob. Appl. 4, 319–329 (1959)
Logan B.F., Shepp L.A.: A variational problem for random Young tableaux. Adv. Math. 26, 206–222 (1977)
Lusztig G.: Characters of Reductive Groups over a Finite Field, volume 107 of Ann. of Math. Studies. Princeton University Press, Princeton (1984)
Macdonald I.G.: Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. 2nd edn. Oxford University Press, New York (1995)
Mathas, A.: Iwahori-Hecke algebras and Schur algebras of the symmetric group, volume 15 of University Lecture Series. Amer. Math. Soc. (1999)
Méliot, P.-L.: Gaussian concentration of the q-characters of the Hecke algebras of type A. (In preparation) (2010)
Okounkov A.: Random matrices and random permutations. Int. Math. Res. Notices. 20, 1043–1095 (2000)
Ram A.: A Frobenius formula for the characters of the Hecke algebras. Invent. Math. 106, 461–488 (1991)
Ram A., Remmel J.B.: Applications of the Frobenius formulas for the characters of the symmetric group and the Hecke algebra of type A. Algebraic Combin. 5, 59–87 (1997)
Ram A., Remmel J.B., Whitehead T.: Combinatorics of the q-basis of symmetric functions. J. Combin. Theory. 76, 231–271 (1996)
The Sage-Combinat community. Sage-Combinat: enhancing sage as a toolbox for computer exploration in algebraic combinatorics (2008) http://combinat.sagemath.org
Śniady P.: Gaussian fluctuations of characters of symmetric groups and of Young diagrams. Probab. Theory Relat. Fields 136, 263–297 (2006)
Stein W.A. et al.: Sage Mathematics Software (Version 4.2+). The Sage Development Team, (2009) http://www.sagemath.org
Strahov E.: A differential model for the deformation of the Plancherel growth process. Adv. Math. 217(6), 2625–2663 (2008)
Thoma E.: Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe. Math. Zeitschrift 85, 40–61 (1964)
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Féray, V., Méliot, PL. Asymptotics of q-plancherel measures. Probab. Theory Relat. Fields 152, 589–624 (2012). https://doi.org/10.1007/s00440-010-0331-6
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DOI: https://doi.org/10.1007/s00440-010-0331-6
Keywords
- Asymptotics of Young diagrams
- Hecke algebras
- Deformation of Plancherel measure
Mathematics Subject Classification (2000)
- 60C05
- 20C30
- 20C08