Skip to main content
SpringerLink
Log in

An introduction to harmonic analysis on the infinite symmetric group

  • Chapter
  • First Online:
Asymptotic Combinatorics with Applications to Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1815))

Abstract

The aim of the present survey paper is to provide an accessible introduction to a new chapter of representation theory—harmonic analysis for noncom- mutative groups with infinite —dimensional dual space.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aldous, D.J.:Exchangeability and related topics.In:Springer Lecture Notes in Math.,1117, 2–199 (1985)

    Google Scholar 

  2. Arratia, R., Barbour, A.D., Tavaré, S.:Poisson processes approximations for the Ewens sampling formula.Ann.Appl.Probab.,2, 519–535 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arratia, R., Barbour, A.D., Tavaré, S.:Random combinatorial structures and prime factorizations.Notices Amer.Math.Soc.,44, no.8, 903–910 (1997)

    MATH  MathSciNet  Google Scholar 

  4. Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations.J.Amer.Math.Soc.,12, no.4, 1119–1178 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Biane, Ph.:Representations of symmetric groups and free probability.Advances in Math.,138,126–181 (1998)

    Google Scholar 

  6. Borodin, A.:Multiplicative central measures on the Schur graph.In:Vershik, A.M.(ed)Representation theory,dynamical systems,combinatorial and algorithmic methods II.Zapiski Nauchnykh Seminarov POMI,240, 44–52 (1997)(Russian);English translation:J.Math.Sci.,96,no.5, 3472-3477 (1999)

    Google Scholar 

  7. Borodin, A.:Characters of symmetric groups and correlation functions of point processes.Funktsional.Anal.Prilozhen.,34, no.1, 12–28 (2000)(Russian);English translation:Funct.Anal.Appl.,34, no.1, 10-23 (2000)

    Google Scholar 

  8. Borodin, A.:Harmonic analysis on the infinite symmetric group and the Whittaker kernel.Algebra Anal.12, no.5,28–63 (2001)(Russian);English translation:St. Petersburg Math.J.,12,no.5, 733-759 (2001)

    Google Scholar 

  9. Borodin, A.:Riemann—Hilbert problem and the discrete Bessel kernel.Intern.Math.Research Notices, no.9,467–494 (2000),math/9912093

    Google Scholar 

  10. Borodin, A.: Discrete gap probabilities and discrete Painlevé equations.Duke Math.J.,to appear;math-ph/0111008

    Google Scholar 

  11. Borodin, A.: Asymptotic representation theory and Riemann-Hilbert problem. In this volume;math/0110318

    Google Scholar 

  12. Borodin, A., Deift, P.: Fredholm determinants, Jimbo-Miwa-Ueno tau-functions,and representation theory.Comm.Pure Appl.Math.,55, no.9,1160–1230 (2002);math-ph/0111007

    Article  MATH  MathSciNet  Google Scholar 

  13. Borodin, A., Okounkov, A., Olshanski, G.: Asymptotics of Plancherel measures for symmetric groups. J.Amer.Math.Soc.,13, 491–515 (2000);math/9905032

    Article  MathSciNet  Google Scholar 

  14. Borodin, A., Olshanski, G.:Point processes and the infinite symmetric group. Math.Research Lett.,5, 799–816 (1998);math/9810015

    MATH  MathSciNet  Google Scholar 

  15. Borodin, A., Olshanski, G.:Distributions on partitions, point processes and the hypergeometric kernel. Comm. Math. Phys.,211, no.2,335–358 (2000); math/9904010

    Article  MATH  MathSciNet  Google Scholar 

  16. Borodin, A., Olshanski, G.:Harmonic functions on multiplicative graphs and interpolation polynomials.Electronic J.Comb.,7 (2000),paper #R28; math/9912124

    Google Scholar 

  17. Borodin, A., Olshanski, G.: Z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and β =2 random matrix ensembles.In: Bleher, P.M., Its, A.R.(eds)Random matrix models and their applications.Mathematical Sciences Research Institute Publications 40, Cambridge Univ.Press,71–94 (2001);math/9905189

    Google Scholar 

  18. Borodin, A., Olshanski, G.: Infinite random matrices and ergodic measures.Comm.Math.Phys.,223, 87–123 (2001);math-ph/0010015

    Article  MATH  MathSciNet  Google Scholar 

  19. Borodin, A., Olshanski, G.: Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes.Ann.Math.,to appear; math/0109194

    Google Scholar 

  20. Borodin, A., Olshanski, G.: Z-measures on partitions and their scaling limits, math-ph/0210148 (2002)

    Google Scholar 

  21. Daley, D.J., Vere-Jones, D.:An introduction to the theory of point processes.Springer series in statistics, Springer,(1988)

    Google Scholar 

  22. Deift, P.: Integrable operators.In:Buslaev, V., Solomyak, M., Yafaev, D.(eds)Differential operators and spectral theory:M.Sh.Birman’ s 70th anniversary collection.American Mathematical Society Translations,ser.2,189, Providence, R.I.,AMS,(1999)

    Google Scholar 

  23. Diaconis, P., Freedman, D.: Partial exchangeability and sufficiency.In: Statistics:Applications and New Directions (Calcutta,1981),Indian Statist.Inst., Calcutta,205–236 (1984)

    Google Scholar 

  24. Dyson, F.J.:Statistical theory of the energy levels of complex systems I,II,III. J.Math.Phys.,3, 140–156, 157-165,166-175 (1962)

    Google Scholar 

  25. Edrei, A.: On the generating functions of totally positive sequences II.J.Analyse Math.,2, 104–109 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ewens, W.J.: The sampling theory of selectively neutral alleles.Theoret.Population Biology,3, 87–112 (1972)

    Article  MathSciNet  Google Scholar 

  27. Ewens, W.J.:Population Genetics Theory-the Past and the Future.In: Lessard, S.(ed)Mathematical and Statistical Developments of Evolutionary Theory. Proc.NATO ASI Symp.,Kluwer, Dordrecht,117–228 (1990)

    Google Scholar 

  28. Griffiths, R.C.:On the distribution of points in a Poisson Dirichlet process. J.Appl.Probab.,25, 336–345 (1988)

    Google Scholar 

  29. Hida, T., Nomoto, H.:Gaussian measure on the projective limit space of spheres. Proc.Japan Academy,40, 31–34 (1964)

    MathSciNet  Google Scholar 

  30. Ignatov, Ts.:On a constant arising in the asymptotic theory of symmetric groups and on Poisson-Dirichlet measures.Teor.Veroyatnost.Primenen.,27, no.1,129–140 (1982) (Russian); English translation:Theory Probab.Appl.,27, 136-147 (1982)

    Google Scholar 

  31. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.:Differential equations for quantum correlation functions.Intern.J.Mod.Phys.,B4,1003–1037 (1990)

    Article  MathSciNet  Google Scholar 

  32. Ivanov, V., Olshanski, G.: Kerov’s central limit theorem for the Plancherel measure on Young diagrams.In:Fomin, S.(ed)Symmetric Functions 2001:Surveys of Developments and Perspectives.NATO Science Series II.Mathematics, Physics and Chemistry,vol.74,Kluwer,93–151 (2001)

    Google Scholar 

  33. Johansson, K.:Discrete orthogonal polynomial ensembles and the Plancherel measure.Ann.Math.(2),153, no.1,259–296 (2001);math/9906120

    Google Scholar 

  34. Kerov, S.V.:Combinatorial examples in the theory of AF-algebras.In:Differential geometry,Lie groups and mechanics X,Zapiski Nauchnykh Seminarov LOMI,172,55–67 (1989)(Russian);English translation:J.Soviet Math.,59, no.5,1063-1071 (1992)

    Google Scholar 

  35. Kerov, S.V.:Generalized Hall-Littlewood symmetric functions and orthogonal polynomials.In:Vershik, A.M.(ed)Representation Theory and Dynamical Systems.Advances in Soviet Math.,Vol.9,Amer.Math.Soc., Providence, R.I., 67–94 (1992)

    Google Scholar 

  36. Kerov, S.V.:Gaussian limit for the Plancherel measure of the symmetric group. Comptes Rendus Acad.Sci.Paris, Série I, 316, 303–308 (1993)

    Google Scholar 

  37. Kerov, S.V.: Subordinators and the actions of permutations with quasi-invariant measure.In: Zapiski Nauchnyh Seminarov POMI,223,181–218 (1995)(Russian);English translation:J.Math.Sci.(New York),87,no.6, 4094-4117 (1997)

    Google Scholar 

  38. Kerov, S.V.: Anisotropic Young diagrams and Jack symmetric functions. Funktsional.Anal.Prilozhen.,34, no.1,51–64 (2000)(Russian);English translation: Funct.Anal.Appl.,34,no.1, 41-51 (2000)

    MathSciNet  Google Scholar 

  39. Kerov, S., Okounkov, A., Olshanski, G.:The boundary of Young graph with Jack edge multiplicities. Intern.Math.Res.Notices,1998:4,173–199 (1998); q-alg/9703037.

    Article  MATH  MathSciNet  Google Scholar 

  40. Kerov, S., Olshanski, G.:Polynomial functions on the set of Young diagrams.Comptes Rendus Acad.Sci.Paris Sér.I,319,121–126 (1994)

    MATH  MathSciNet  Google Scholar 

  41. Kerov, S., Olshanski, G., Vershik, A.:Harmonic analysis on the infinite symmetric group.A deformation of the regular representation.Comptes Rend.Acad. Sci.Paris, Sér.I, 316,773–778 (1993);detailed version in preparation

    MATH  MathSciNet  Google Scholar 

  42. Kerov, S.V., Tsilevich, N.V.: Stick breaking process generates virtual permutations with Ewens distribution In: Zapiski Nauchnyh Seminarov POMI,223, 162–180 (1995)(Russian);English translation:J.Math.Sci.(New York),87, no.6,4082-4093 (1997)

    Google Scholar 

  43. Kingman, J.F.C.:Random discrete distributions.J.Royal Statist.Soc.B,37,1–22 (1975)

    Google Scholar 

  44. Kingman, J.F.C.: The population structure associated with the Ewens sampling formula.Theoret.Population Biology,11, 274–283 (1977)

    Article  MathSciNet  Google Scholar 

  45. Kingman, J.F.C.:Random partitions in population genetics.Proc.Roy.Soc. London A.,361,1–20 (1978)

    Google Scholar 

  46. Kingman, J.F.C.: The representation of partition structures.J.London Math. Soc.(2),18,374–380 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  47. Kingman, J.F.C.: Poisson processes.Oxford University Press (1993)

    Google Scholar 

  48. Lenard, A.:Correlation functions and the uniqueness of the state in classical statistical mechanics.Comm.Math.Phys,30,35–44 (1973)

    Google Scholar 

  49. Macchi, O.:The coincidence approach to stochastic point processes.Adv.Appl. Prob.,7,83–122 (1975)

    Google Scholar 

  50. Macchi, O.:The fermion process—a model of stochastic point process with repulsive points.In:Transactions of the Seventh Prague Conference on Information Theory,Statistical Decision Functions,Random Processes and of the Eighth European Meeting of Statisticians (Tech.Univ.Prague, Prague,1974), Vol.A,Reidel, Dordrecht, 391–398 (1977)

    Google Scholar 

  51. Murray, F.J., von Neumann, J.:On rings of operators IV.Ann.Math.44,716–808 (1943)

    Article  Google Scholar 

  52. Mehta, M.L.:Random matrices,2nd edition.Academic Press, New York (1991)

    Google Scholar 

  53. Neretin, Yu.A.:Hua type integrals over unitary groups and over projective limits of unitary groups.Duke Math.J.,114,239–266 (2002);math-ph/0010014

    Google Scholar 

  54. Nagao, T., Wadati, M.: Correlation functions of random matrix ensembles related to classical orthogonal polynomials.J.Phys.Soc.Japan,60,no.10,3298–3322 (1991)

    Article  MathSciNet  Google Scholar 

  55. Naimark, M.A.: Normed rings.Nauka,Moscow (1962)(Russian);English translation:Normed algebras.Wolters-Noordho-,Groningen,(1972)

    Google Scholar 

  56. Okounkov, A.:Thoma’s theorem and representations of infinite bisymmetric group.Funktsion.Anal.Prilozhen.,28,no.2,31–40 (1994)(Russian);English translation:Funct.Anal.Appl.,28,no.2,101-107 (1994)

    Google Scholar 

  57. Okounkov, A.Yu.:On representations of the infinite symmetric group.In:Vershik, A.M.(ed)Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods II,Zap.Nauchn.Semin.POMI,240,167–229 (1997)(Russian);English translation:J.Math.Sci.(New York),96,no.5, 3550-3589 (1999)

    Google Scholar 

  58. Okounkov, A.: SL (2) and z-measures.In:Bleher, P.M., Its, A.R.(eds)Random matrix models and their applications.Mathematical Sciences Research Institute Publications,40,Cambridge Univ.Press,407–420 (2001);math/0002136

    Google Scholar 

  59. Okounkov, A.:Infinite wedge and measures on partitions.Selecta Math.(New Ser.),7,57–81 (2001);math/9907127

    Google Scholar 

  60. Okounkov, A., Olshanski, G.:Shifted Schur functions.Algebra i Analiz,9, no.2, 73–146 (1997)(Russian);English translation:St. Petersburg Math.J.,9,no.2, 239-300 (1998)

    Google Scholar 

  61. Olshanski, G.:Unitary representations of infinite-dimensional pairs (G,K) and the formalism of R.Howe.Doklady AN SSSR,269,33–36(1983)(Russian); English translation:Soviet Math.Doklady,27,no.2, 290-294 (1983)

    Google Scholar 

  62. Olshanski, G.: Unitary representations of infinite—dimensional pairs (G,K)and the formalism of R.Howe.In: Vershik, A., Zhelobenko, D.(eds)Representation of Lie Groups and Related Topics.Advanced Studies in Contemporary Math., 7, Gordon and Breach Science Publishers, New York etc., 269–463 (1990)

    Google Scholar 

  63. Olshanski, G.:Unitary representations of (G,K)-pairs connected with the infinite symmetric group S (∞).Algebra i Analiz,1, no.4,178–209 (1989)(Russian);English translation:Leningrad Math.J.,1,983-1014 (1990)

    Google Scholar 

  64. Olshanski, G.:The problem of harmonic analysis on the infinite—dimensional unitary group.J.Funct.Anal.,to appear;math/0109193

    Google Scholar 

  65. Olshanski, G., Regev, A., Vershik, A.:Frobenius—Schur functions.In: Joseph, A., Melnikov, A., Rentschler, R.(eds)Studies in Memory of Issai Schur.Birkhäuser, to appear;math/0110077.

    Google Scholar 

  66. Olshanski, G.:Point processes and the infinite symmetric group.Part I:The general formalism and the density function.math/9804086 (1998)

    Google Scholar 

  67. Borodin, A.:Point processes and the infinite symmetric group.Part II:Higher correlation functions.math/9804087 (1998)

    Google Scholar 

  68. Borodin, A., Olshanski, G.:Point processes and the infinite symmetric group.Part III:Fermion point processes.math/9804088 (1998)

    Google Scholar 

  69. Borodin, A.:Point processes and the infinite symmetric group.Part IV:Matrix Whittaker kernel.math/9810013 (1998)

    Google Scholar 

  70. Olshanski, G.:Point processes and the infinite symmetric group.Part V: Analysis of the matrix Whittaker kernel.math/9810014 (1998)

    Google Scholar 

  71. Pickrell, D.:Measures on infinite dimensional Grassmann manifold.J.Funct. Anal.,70,323–356 (1987)

    Google Scholar 

  72. Rozhkovskaya, N.A.:Multiplicative distributions on Young graph.In:Vershik, A.M.(ed)Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods II.Zapiski Nauchnykh Seminarov POMI,240,Nauka, St. Petersburg, 246–257 (1997)(Russian);English translation:J.Math.Sci.(New York),96,no.5, 3600-3608 (1999)

    Google Scholar 

  73. Shimomura, H.:On the construction of invariant measure over the orthogonal group on the Hilbert space by the method of Cayley transformation.Publ.RIMS Kyoto Univ.,10, 413–424 (1974/75)

    Google Scholar 

  74. Soshnikov, A.: Determinantal random point fields.Uspekhi Mat.Nauk,55, no.5,107–160 (2000)(Russian);English translation:Russian Math.Surveys, 55, no.5, 923-975 (2000);math/0002099

    MathSciNet  Google Scholar 

  75. Stratila, S., Voiculescu, D.:Representations of AF-algebras and of the group U(∞).Springer Lecture Notes,486(1975)

    Google Scholar 

  76. Thoma, E.:Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen,symmetrischen Gruppe.Math.Zeitschr.,85,40–61(1964)

    Google Scholar 

  77. Thoma, E.:Characters of infinite groups.In:Arsene, Gr., Strătilă, S., Verona, A., Voiculescu, D.(eds)Operator Algebras and Group Representations,Vol.2, Pitman,23–32 (1984)

    Google Scholar 

  78. Tracy, C.A., Widom, H.:Universality of distribution functions of random matrix theory II.In:Harnad, J., Sabidussi, G., Winternitz, P.(eds)Integrable Systems: From Classical to Quantum.CRM Proceedings & Lecture Notes,26, Amer.Math.Soc.,Providence, 251–264 (2000)

    Google Scholar 

  79. Vershik, A.M.:Description of invariant measures for the actions of some infinite-dimensional groups.Doklady AN SSSR,218,749–752 (1974)(Russian);English translation:Soviet Math.Doklady,15,1396-1400 (1974)

    Google Scholar 

  80. Vershik, A.M.:Asymptotic distribution of decompositions of natural numbers into prime divisors.Dokl.Akad.Nauk SSSR,289, no.2,269–272 (1986)(Russian);English translation:Soviet Math.Doklady,34,57-61 (1987)

    Google Scholar 

  81. Vershik, A.M.:Statistical mechanics of combinatorial partitions,and their limit shapes.Funktsional.Anal.Prilozhen.,30,no.2,19–39 (1996)(Russian);English translation:Funct.Anal.Appl.,30,90-105 (1996)

    Google Scholar 

  82. Vershik, A.M., Kerov, S.V.:Characters and factor representations of the infinite symmetric group.Doklady AN SSSR,257,1037–1040 (1981)(Russian); English translation:Soviet Math.Doklady,23,389-392 (1981)

    MathSciNet  Google Scholar 

  83. Vershik, A.M., Kerov, S.V.:Asymptotic theory of characters of the symmetric group.Funktsion.Anal.Prilozhen.,15,no.4,15–27 (1981)(Russian);English translation:Funct.Anal.Appl.,15, no.4, 246-255 (1981)

    MathSciNet  Google Scholar 

  84. Vershik, A.M., Kerov, S.V.: Characters and factor representations of the infinite unitary group.Doklady AN SSSR,267,no.2, 272–276 (1982)(Russian); English translation:Soviet Math.Doklady,26,570-574 (1982)

    MathSciNet  Google Scholar 

  85. Vershik, A.M., Kerov, S.V.: Locally semisimple algebras.Combinatorial theory and the K 0 functor.In: Itogi Nauki,Sovr.Probl.Mat.,Noveish.Dostizh., VINITI,26,3–56 (1985)(Russian);English translation:J.Soviet Math.,38,1701-1733 (1987)

    MathSciNet  Google Scholar 

  86. Schmidt, A., Vershik, A.: Limit measures that arise in the asymptotic theory of symmetric groups.I,II.Teor.Verojatn. i Prim.,22,no.1,72–88 (1977);23, no.1, 42-54 (1978)(Russian);English translation:Theory of Probab.Appl., 22,70-85 (1977); 23,36-49 (1978)

    Google Scholar 

  87. Voiculescu, D.: Représentations factorielles de type II1 de U (∞).J.Math.Pures Appl.,55,1–20 (1976)

    MATH  MathSciNet  Google Scholar 

  88. Wassermann, A.J.: Automorphic actions of compact groups on operator algebras.Thesis,University of Pennsylvania (1981)

    Google Scholar 

  89. Watterson, G.A.:The stationary distribution of the infinitely many-alleles diffusion model.J.Appl.Probab.,13,639–651 (1976)

    Google Scholar 

  90. Yamasaki, Y.:Projective limit of Haar measures on O (n).Publ.RIMS, KyotoUniv.,8,141–149 (1972/73)

    Google Scholar 

  91. Yamasaki, Y.: Kolmogorov’s extension theorem for infinite measure.Publ. RIMS, Kyoto Univ.,10,381–411 (1974/75)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dobrushin Mathematics Laboratory, Institute for Information Transmission Problems, Bolshoy Karetny 19, Moscow, 101447, 101447, Russia

    Grigori Olshanski

Authors
  1. Grigori Olshanski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. St. Petersburg Department of the Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

    Anatoly M. Vershik

  2. St. Petersburg Department of the Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

    Yuri Yakubovich

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Olshanski, G. (2003). An introduction to harmonic analysis on the infinite symmetric group. In: Vershik, A.M., Yakubovich, Y. (eds) Asymptotic Combinatorics with Applications to Mathematical Physics. Lecture Notes in Mathematics(), vol 1815. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44890-X_6

Download citation

  • DOI: https://doi.org/10.1007/3-540-44890-X_6

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40312-8

  • Online ISBN: 978-3-540-44890-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation