Inventiones mathematicae

, Volume 158, Issue 3, pp 551–642

# Harmonic analysis on the infinite symmetric group

• Sergei Kerov
• Grigori Olshanski
• Anatoly Vershik
Article

## Abstract

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.

## Keywords

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.

## References

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)
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)
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)
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)
8. BO4.
Borodin, A., Olshanski, G.: Infinite random matrices and ergodic measures. Commun. Math. Phys. 223, 87–123 (2001)
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)
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)
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)
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)
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)
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)
25. Kin1.
Kingman, J.F.C.: The population structure associated with the Ewens sampling formula. Theor. Popul. Biol. 11, 274–283 (1977)
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)
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)
33. Ner3.
Neretin, Yu.A.: Plancherel formula for Berezin deformation of L2 on Riemannian symmetric space. J. Funct. Anal. 189, 336–408 (2002)
34. Ok1.
Okounkov, A.Yu.: Thoma’s theorem and representations of infinite bisymmetric group. Funct. Anal. Appl. 28, 101–107 (1994)
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)
38. OkV.
Okounkov, A., Vershik, A.: A new approach to representation theory of symmetric groups. Sel. Math., New Ser. 2, 581–605 (1996)
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)
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)
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)
48. Pit.
Pitman, J.: Combinatorial stochastic processes. Lecture Notes for St. Flour Summer School, July 2002, available via http://stat-www.berkeley.edu/users/pitman/Google 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)
51. Shir.
Shiryaev, A.: Probability. New York: Springer 1996Google Scholar
52. Sta.
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)
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)
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)
59. VK2.
Vershik, A.M., Kerov, S.V.: Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl. 15, 246–255 (1981)
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)
62. Was.
Wassermann, A.J.: Automorphic actions of compact groups on operator algebras. Thesis, University of Pennsylvania 1981Google Scholar