Inventiones mathematicae

, Volume 158, Issue 3, pp 551–642 | Cite as

Harmonic analysis on the infinite symmetric group

  • Sergei Kerov
  • Grigori Olshanski
  • Anatoly Vershik


The infinite symmetric group S(∞), whose elements are finite permutations of {1,2,3,...}, is a model example of a “big” group. By virtue of an old result of Murray–von Neumann, the one–sided regular representation of S(∞) in the Hilbert space ℓ2(S(∞)) generates a type II1 von Neumann factor while the two–sided regular representation is irreducible. This shows that the conventional scheme of harmonic analysis is not applicable to S(∞): for the former representation, decomposition into irreducibles is highly non–unique, and for the latter representation, there is no need of any decomposition at all. We start with constructing a compactification \(\mathfrak{S}\supset{S(\infty)}\), which we call the space of virtual permutations. Although \(\mathfrak{S}\) is no longer a group, it still admits a natural two–sided action of S(∞). Thus, \(\mathfrak{S}\) is a G–space, where G stands for the product of two copies of S(∞). On \(\mathfrak{S}\), there exists a unique G-invariant probability measure μ1, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ1 into a family {μ t : t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {T z : z∈ℂ} of unitary representations of G, called generalized regular representations (each representation T z with z≠=0 can be realized in the Hilbert space \(L^2(\mathfrak{S}, \mu_t)\), where t=|z|2). As |z|→∞, the generalized regular representations T z approach, in a suitable sense, the “naive” two–sided regular representation of the group G in the space ℓ2(S(∞)). In contrast with the latter representation, the generalized regular representations T z are highly reducible and have a rich structure. We prove that any T z admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z1, z2, the spectral types of the representations \(T_{z_1}\) and \(T_{z_2}\) are shown to be disjoint. In the case z∈ℤ, a complete description of the spectral type is obtained. Further work on the case z∈ℂ∖ℤ reveals a remarkable link with stochastic point processes and random matrix theory.


Symmetric Group Unitary Representation Young Diagram Spectral Type Inductive Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ald.
    Aldous, D.J.: Exchangeability and related topics. Lect. Notes Math. 1117, pp. 2–199. Springer 1985Google Scholar
  2. BR.
    Berele, A., Regev, A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. Adv. Math. 64, 118–175 (1987)MathSciNetCrossRefGoogle Scholar
  3. Bor1.
    Borodin, A.M.: Multiplicative central measures on the Schur graph. In: Representation theory, dynamical systems, combinatorial and algorithmic methods II, ed. by A.M. Vershik. Zap. Nauchn. Semin. POMI, vol. 240, pp. 44–52. St. Petersburg: Nauka 1997 (Russian); English transl. in J. Math. Sci., New York 96, 3472–3477 (1999)CrossRefGoogle Scholar
  4. Bor2.
    Borodin, A.: Harmonic analysis on the infinite symmetric group and the Whittaker kernel. Algebra Anal. 12, 28–63 (2000) (Russian); English translation: St. Petersbg. Math. J. 12, 733–759 (2001)Google Scholar
  5. BO1.
    Borodin, A., Olshanski, G.: Point processes and the infinite symmetric group. Math. Res. Lett. 5, 799–816 (1998)MathSciNetCrossRefGoogle Scholar
  6. BO2.
    Borodin, A., Olshanski, G.: Harmonic functions on multiplicative graphs and interpolation polynomials. Electron. J. Comb. 7 (2000), paper #R28Google Scholar
  7. BO3.
    Borodin, A., Olshanski, G.: Distributions on partitions, point processes, and the hypergeometric kernel. Commun. Math. Phys. 211, 335–358 (2000)CrossRefGoogle Scholar
  8. BO4.
    Borodin, A., Olshanski, G.: Infinite random matrices and ergodic measures. Commun. Math. Phys. 223, 87–123 (2001)CrossRefGoogle Scholar
  9. BO5.
    Borodin, A., Olshanski, G.: Harmonic analysis on the infinite–dimensional unitary group and determinantal point processes. arXiv:math/0109194. To appear in Ann. Math.Google Scholar
  10. CFS.
    Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic theory. Moscow: Nauka 1980 (Russian); English translation: New York: Springer 1982Google Scholar
  11. DF.
    Diaconis, P., Freedman, D.: Partial exchangeability and sufficiency. In: Statistics: Applications and New Directions (Calcutta, 1981), pp. 205–236. Calcutta: Indian Statist. Inst. 1984Google Scholar
  12. Dix.
    Dixmier, J.: Les C*–algèbres et leurs représentations. Paris: Gauthier–Villars 1969Google Scholar
  13. Edr.
    Edrei, A.: On the generating functions of totally positive sequences II. J. Anal. Math. 2, 104–109 (1952)MathSciNetCrossRefGoogle Scholar
  14. GV.
    Gelfand, I.M., Vilenkin, N.Ya.: Some applications of harmonic analysis. Rigged Hilbert spaces (Generalized functions 4). Moscow: Fizmatgiz 1961 (Russian); English translation: Applications of harmonic analysis. New York, London: Academic Press 1964Google Scholar
  15. Ism.
    Ismagilov, R.S.: Representations of infinite–dimensional groups. Transl. Math. Monogr. 152. Providence, RI: Amer. Math. Soc. 1996Google Scholar
  16. Ka.
    Kaimanovich, V.A.: Measure–theoretic boundaries of Markov chains, 0-2 laws and entropy. In: Harmonic Analysis and Discrete Potential Theory, ed. by M.A. Picardello, pp. 145–180. Frascati 1991. New York: Plenum 1992Google Scholar
  17. Kak.
    Kakutani, S.: On equivalence of infinite product measures. Ann. Math. 49, 214–224 (1948)MathSciNetCrossRefGoogle Scholar
  18. Ker1.
    Kerov, S.V.: Subordinators and the actions of permutations with quasi–invariant measure. Zap. Nauchn. Semin. POMI, vol. 223, 181–218 (1995) (Russian); English translation: J. Math. Sci., New York 87, 4094–4117 (1997)Google Scholar
  19. Ker2.
    Kerov, S.V.: The boundary of Young lattice and random Young tableaux. In: Formal power series and algebraic combinatorics. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 24, pp. 133–158. Providence, RI: Am. Math. Soc. 1996Google Scholar
  20. Ker3.
    Kerov, S.V.: Anisotropic Young diagrams and Jack symmetric functions. Funkts. Anal. Prilozhen. 34, 51–64 (2000) (Russian); English translation: Funct. Anal. Appl. 34, 41–51 (2000)MathSciNetCrossRefGoogle Scholar
  21. Ker4.
    Kerov, S.V.: Asymptotic representation theory of the symmetric group and its applications in analysis, 201 pp. Providence, RI: Am. Math. Soc. 2003Google Scholar
  22. KOO.
    Kerov, S., Okounkov, A., Olshanski, G.: The boundary of Young graph with Jack edge multiplicities. Int. Math. Res. Not. 4, 173–199 (1998)MathSciNetCrossRefGoogle Scholar
  23. KO.
    Kerov, S., Olshanski, G.: Polynomial functions on the set of Young diagrams. C. R. Acad. Sci., Paris, Sér. I, Math. 319, 121–126 (1994)MathSciNetGoogle Scholar
  24. KOV.
    Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group. A deformation of the regular representation. C. R. Acad. Sci., Paris, Sér. I, Math. 316, 773–778 (1993)MathSciNetGoogle Scholar
  25. Kin1.
    Kingman, J.F.C.: The population structure associated with the Ewens sampling formula. Theor. Popul. Biol. 11, 274–283 (1977)MathSciNetCrossRefGoogle Scholar
  26. Kin2.
    Kingman, J.F.C.: Poisson processes. Oxford: Oxford University Press 1993Google Scholar
  27. Kir.
    Kirillov, A.A.: Elements of the theory of representations. Grundlehren der mathematischen Wissenschaften 220. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  28. Mac.
    Macdonald, I.G.: Symmetric functions and Hall polynomials, 2nd edition. Oxford: Oxford University Press 1995Google Scholar
  29. MvN.
    Murray, F.J., von Neumann, J.: On rings of operators IV. Ann. Math. 44, 716–808 (1943)CrossRefGoogle Scholar
  30. Nai.
    Naimark, M.A.: Normed rings. Translated from the first Russian edition. Groningen, The Netherlands: Wolters-Noordhoff Publishing 1970Google Scholar
  31. Ner1.
    Neretin, Yu.A.: Categories of symmetries and infinite–dimensional groups. Lond. Math. Soc. Monogr., New Ser. 16. Oxford: Oxford Univ. Press 1996; Russian edition: Moscow: URSS 1998Google Scholar
  32. Ner2.
    Neretin, Yu.A.: Hua type integrals over unitary groups and over projective limits of unitary groups. Duke Math. J. 114, 239–266 (2002)MathSciNetCrossRefGoogle Scholar
  33. Ner3.
    Neretin, Yu.A.: Plancherel formula for Berezin deformation of L2 on Riemannian symmetric space. J. Funct. Anal. 189, 336–408 (2002)MathSciNetCrossRefGoogle Scholar
  34. Ok1.
    Okounkov, A.Yu.: Thoma’s theorem and representations of infinite bisymmetric group. Funct. Anal. Appl. 28, 101–107 (1994)MathSciNetCrossRefGoogle Scholar
  35. Ok2.
    Okounkov, A.Yu.: On representations of the infinite symmetric group. Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods II, ed. by A.M. Vershik. Zap. Nauchn. Semin. POMI, vol. 240, pp. 167–229 (1997) (Russian); English translation in J. Math. Sci., New York 96, 3550–3589 (1999)Google Scholar
  36. Ok3.
    Okounkov, A.: SL(2) and z–measures. In: Random matrix models and their applications, ed. by P.M. Bleher, A.R. Its. Math. Sci. Res. Inst. Publ. 40, pp. 407–420. Cambridge: Cambridge Univ. Press 2001Google Scholar
  37. OkOl.
    Okounkov, A., Olshanski, G.: Shifted Schur functions. Algebra Anal. 9, 73–146 (1997) (Russian); English translation: St. Petersburg Math. J. 9, 239–300 (1998)MathSciNetGoogle Scholar
  38. OkV.
    Okounkov, A., Vershik, A.: A new approach to representation theory of symmetric groups. Sel. Math., New Ser. 2, 581–605 (1996)MathSciNetCrossRefGoogle Scholar
  39. Ol1.
    Olshanski, G.: Unitary representations of the infinite–dimensional classical groups U(p,∞), SO(p,∞), Sp(p,∞) and the corresponding motion groups. Funkts. Anal. Prilozh. 12, 20–44 (1978) (Russian); English translation: Funct. Anal. Appl. 12, 185–195 (1979)CrossRefGoogle Scholar
  40. Ol2.
    Olshanski, G.: Unitary representations of infinite-dimensional pairs (G,K) and the formalism of R. Howe. Soviet Math. Doklady 27, 290–294 (1983)Google Scholar
  41. Ol3.
    Olshanski, G.: Unitary representations of (G,K)-pairs connected with the infinite symmetric group S(∞). Algebra Anal. 1, 178–209 (1989) (Russian); English translation: Leningrad Math. J. 1, 983–1014 (1990)Google Scholar
  42. Ol4.
    Olshanski, G.: Unitary representations of infinite-dimensional pairs (G,K) and the formalism of R. Howe. In: Representation of Lie Groups and Related Topics, ed. by A. Vershik, D. Zhelobenko, pp. 269–463. Adv. Stud. Contemp. Math. 7. New York etc.: Gordon and Breach Science Publishers 1990Google Scholar
  43. Ol5.
    Olshanski, G.: Point processes related to the infinite symmetric group. In: The orbit method in geometry and physics: in honor of A.A. Kirillov, ed. by Ch. Duval, L. Guieu, V. Ovsienko, pp. 349–393. Prog. Math. 213. Boston, MA: Birkhäuser 2003Google Scholar
  44. Ol6.
    Olshanski, G.: An introduction to harmonic analysis on the infinite symmetric group. In: Asymptotic Combinatorics with Applications to Mathematical Physics, ed. by A. Vershik. Lect. Notes Math. 1815. Springer 2003Google Scholar
  45. Ol7.
    Olshanski, G.: The problem of harmonic analysis on the infinite–dimensional unitary group. J. Funct. Anal. 205, 464–524 (2003)MathSciNetCrossRefGoogle Scholar
  46. ORV.
    Olshanski, G., Regev, A., Vershik, A.: Frobenius–Schur functions. In: Studies in Memory of Issai Schur, ed. by A. Joseph, A. Melnikov, R. Rentschler, pp. 251–300. Prog. Math. 210, Birkhäuser 2003Google Scholar
  47. Pic.
    Pickrell, D.: Measures on infinite dimensional Grassmann manifold. J. Funct. Anal. 70, 323–356 (1987)MathSciNetCrossRefGoogle Scholar
  48. Pit.
    Pitman, J.: Combinatorial stochastic processes. Lecture Notes for St. Flour Summer School, July 2002, available via Scholar
  49. Rev.
    Revuz, D.: Markov chains. Amsterdam: North–Holland 1984Google Scholar
  50. Rozh.
    Rozhkovskaya, N.A.: Multiplicative distributions on Young graph. Representation theory, dynamical systems, combinatorial and algorithmic methods II, ed. by A.M. Vershik. Zap. Nauchn. Semin. POMI, vol. 240, pp. 246–257. St. Petersburg: Nauka 1997 (Russian); English translation: J. Math. Sci., New York 96, 3600–3608 (1999)MathSciNetCrossRefGoogle Scholar
  51. Shir.
    Shiryaev, A.: Probability. New York: Springer 1996Google Scholar
  52. Sta.
    Stanley, R.P.: Enumerative combinatorics. Wadsworth, Inc. 1986Google Scholar
  53. SV.
    Strătilă, S., Voiculescu, D.: Representations of AF–algebras and of the group U(∞). Lect. Notes Math. 486. Springer 1975Google Scholar
  54. TE.
    Tavaré, S., Ewens, W.J.: The Ewens Sampling Formula. In: Encyclopedia of Statistical Sciences, vol. 2, ed. by S. Kotz, C.B. Read, D.L. Banks, pp. 230–234. New York: Wiley 1998Google Scholar
  55. Tho1.
    Thoma, E.: Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Z. 85, 40–61 (1964)MathSciNetCrossRefGoogle Scholar
  56. Tho2.
    Thoma, E.: Characters of infinite groups. In: Operator algebras and group representations, vol. 2, ed. by Gr. Arsene, S. Strătilă, A. Verona, D. Voiculescu, pp. 23–32. Pitman 1984Google Scholar
  57. Ver.
    Vershik, A.: Description of invariant measures for the actions of some infinite-dimensional groups. Dokl. Akad. Nauk SSSR 218, 749–752 (1974) (Russian); English translation in Soviet Math. Doklady 15, 1396–1400 (1974)MathSciNetGoogle Scholar
  58. VK1.
    Vershik, A.M., Kerov, S.V.: Characters and factor representations of the infinite symmetric group. Dokl. Akad. Nauk SSSR 257, 1037–1040 (1981) (Russian); English translation in Soviet Math. Doklady 23, 389–392 (1981)MathSciNetGoogle Scholar
  59. VK2.
    Vershik, A.M., Kerov, S.V.: Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl. 15, 246–255 (1981)MathSciNetCrossRefGoogle Scholar
  60. VK3.
    Vershik, A.M., Kerov, S.V.: The Grothendieck group of the infinite symmetric group and symmetric functions with the elements of the K0-functor theory of AF-algebras. Representation of Lie groups and related topics, ed. by A.M. Vershik, D.P. Zhelobenko. Adv. Stud. Contemp. Math. 7, pp. 36–114. Gordon and Breach 1990Google Scholar
  61. Voi.
    Voiculescu, D.: Représentations factorielles de type II1 de U(∞). J. Math. Pures Appl. 55, 1–20 (1976)MathSciNetGoogle Scholar
  62. Was.
    Wassermann, A.J.: Automorphic actions of compact groups on operator algebras. Thesis, University of Pennsylvania 1981Google Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsMoscowRussia
  2. 2.Steklov Institute of Mathematics at St. PetersburgSt. PetersburgRussia

Personalised recommendations