Skip to main content
SpringerLink
Log in

Three Theorems on the Uniqueness of the Plancherel Measure from Different Viewpoints

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We consider three uniqueness theorems: one from the theory of meromorphic functions, another from asymptotic combinatorics, and the third concerns representations of the infinite symmetric group. The first theorem establishes the uniqueness of the function exp z in a class of entire functions. The second is about the uniqueness of a random monotone nonde-generate numbering of the two-dimensional lattice ℤ 2+ , or of a nondegenerate central measure on the space of infinite Young tableaux. And the third theorem establishes the uniqueness of a representation of the infinite symmetric group \(\mathfrak{S}_{\mathbb{N}}\) whose restrictions to finite subgroups have vanishingly few invariant vectors. However, in fact all the three theorems are, up to a nontrivial rephrasing of conditions from one area of mathematics in terms of another area, the same theorem! Up to now, researchers working in each of these areas have not been aware of this equivalence. The parallelism of these uniqueness theorems on the one hand and the difference of their proofs on the other call for a deeper analysis of the nature of uniqueness and suggest transferring the methods of proof between the areas. More exactly, each of these theorems establishes the uniqueness of the so-called Plancherel measure, which is the main object of our paper. In particular, we show that this notion is general for all locally finite groups.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Aissen, I. J. Schoenberg, and A. M. Whitney, “On the generating functions of totally positive sequences. I,” J. Anal. Math. 2, 93–103 (1952).

    Article  MathSciNet  Google Scholar 

  2. I. N. Bernshtein, I. M. Gel’fand, and S. I. Gel’fand, “Models of representations of compact Lie groups,” Funct. Anal. Appl. 9(4), 322–324 (1975) [transl. from Funkts. Anal. Prilozh. 9 (4), 61–62 (1975)].

    Article  MathSciNet  Google Scholar 

  3. A. Borodin, A. Okounkov, and G. Olshanski, “Asymptotics of Plancherel measures for symmetric groups,” J. Am. Math. Soc. 13(3), 481–515 (2000).

    Article  MathSciNet  Google Scholar 

  4. V. M. Buchstaber and A. A. Glutsyuk, “Total positivity, Grassmannian and modified Bessel functions,” in Functional Analysis and Geometry: Selim Grigorievich Krein Centennial (Am. Math. Soc., Providence, RI, 2019), Contemp. Math. 733, pp. 97–107; arXiv: 1708.02154 [math.DS].

    Chapter  Google Scholar 

  5. A. Bufetov and V. Gorin, “Stochastic monotonicity in Young graph and Thoma theorem,” Int. Math. Res. Not. 2015(23), 12920–12940 (2015).

    MathSciNet  MATH  Google Scholar 

  6. J. Dixmier, Les C*-algèbres et leurs represéntations (Gauthier-Villars, Paris, 1969).

    MATH  Google Scholar 

  7. A. Edrei, “On the generating functions of totally positive sequences. II,” J. Anal. Math. 2, 104–109 (1952).

    Article  MathSciNet  Google Scholar 

  8. S. Fomin, “Duality of graded graphs,” J. Algebr. Comb. 3(4), 357–404 (1994).

    Article  MathSciNet  Google Scholar 

  9. S. Fomin and A. Zelevinsky, “Total positivity: Tests and parametrizations,” Math. Intell. 22(1), 23–33 (2000); arXiv: math/9912128 [math.RA].

    Article  MathSciNet  Google Scholar 

  10. F. R. Gantmacher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, rev. ed. (AMS Chelsea Publ., Providence, RI, 2002).

    Book  Google Scholar 

  11. S. Karlin, Total Positivity (Stanford Univ. Press, Stanford, CA, 1968), Vol. 1.

    MATH  Google Scholar 

  12. S. V. Kerov, “Generalized Hall–Littlewood symmetric functions and orthogonal polynomials,” in Representation Theory and Dynamical Systems (Am. Math. Soc., Providence, RI, 1992), Adv. Sov. Math. 9, pp. 67–94.

    Chapter  Google Scholar 

  13. S. V. Kerov, Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis (Am. Math. Soc., Providence, RI, 2003), Transl. Math. Monogr. 219.

    Book  Google Scholar 

  14. A. A. Klyachko, “Centralizers of involutions and models of the symmetric and full linear groups,” in Studies in Number Theory (Saratov. Gos. Univ., Saratov, 1978), Vol. 7, pp. 59–64 [in Russian].

    Google Scholar 

  15. B. F. Logan and L. A. Shepp, “A variational problem for random Young tableaux,” Adv. Math. 26(2), 206–222 (1977).

    Article  MathSciNet  Google Scholar 

  16. G. Lusztig, “A survey of total positivity,” Milan J. Math. 76, 125–134 (2008); arXiv: 0705.3842 [math.RT].

    Article  MathSciNet  Google Scholar 

  17. K. Matveev, “Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture,” Ann. Math., Ser. 2, 189(1), 277–316 (2019).

    Article  MathSciNet  Google Scholar 

  18. A. Yu. Okounkov, “Thoma’s theorem and representations of the infinite bisymmetric group,” Funct. Anal. Appl. 28(2), 100–107 (1994) [transl. from Funkts. Anal. Prilozh. 28 (2), 31–40 (1994)].

    Article  MathSciNet  Google Scholar 

  19. A. Okounkov and G. Olshanski, “Shifted Schur functions,” St. Petersburg Math. J. 9(2), 239–300 (1998) [transl. from Algebra Anal. 9 (2), 73–146 (1997)].

    MathSciNet  MATH  Google Scholar 

  20. F. V. Petrov, “The asymptotics of traces of paths in the Young and Schur graphs,” Zap. Nauchn. Semin. POMI 468, 126–137 (2018).

    Google Scholar 

  21. D. Romik and P. Śniady, “Jeu de taquin dynamics on infinite Young tableaux and second class particles,” Ann. Probab. 43(2), 682–737 (2015).

    Article  MathSciNet  Google Scholar 

  22. I. J. Schoenberg, “Some analytical aspects of the problem of smoothing,” in Studies and Essays: Courant Anniversary Volume (Interscience, New York, 1948), pp. 351–370.

    Google Scholar 

  23. J.-P. Serre, Représentations linéaires des groupes finis (Hermann, Paris, 1967).

    MATH  Google Scholar 

  24. P. Śniady, “Robinson–Schensted–Knuth algorithm, jeu de taquin, and Kerov–Vershik measures on infinite tableaux,” SIAM J. Discrete Math. 28(2), 598–630 (2014).

    Article  MathSciNet  Google Scholar 

  25. R. P. Stanley, “Differential posets,” J. Am. Math. Soc. 1(4), 919–961 (1988).

    Article  MathSciNet  Google Scholar 

  26. E. Thoma, “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe,” Math. Z. 85, 40–61 (1964).

    Article  MathSciNet  Google Scholar 

  27. A. M. Vershik, “A statistical sum associated with Young diagrams,” J. Sov. Math. 47(2), 2379–2386 (1989) [transl. from Zap. Nauchn. Semin. LOMI 164, 20–29 (1987)].

    Article  Google Scholar 

  28. A. M. Vershik, “Asymptotic combinatorics and algebraic analysis,” in Proc. Int. Congr. Math., Zürich, 1994 (Birkhäuser, Basel, 1995), Vol. II, pp. 1384–1394.

    Google Scholar 

  29. A. M. Vershik, “Statistical mechanics of combinatorial partitions, and their limit shapes,” Funct. Anal. Appl. 30(2), 90–105 (1996) [transl. from Funkts. Anal. Prilozh. 30 (2), 19–30 (1996)].

    Article  MathSciNet  Google Scholar 

  30. A. M. Vershik, “The problem of describing central measures on the path spaces of graded graphs,” Funct. Anal. Appl. 48(4), 256–271 (2014) [transl. from Funkts. Anal. Prilozh. 48 (4), 26–46 (2014)].

    Article  MathSciNet  Google Scholar 

  31. A. M. Vershik and S. V. Kerov, “Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables,” Sov. Math. Dokl. 18, 527–531 (1977) [transl. from Dokl. Akad. Nauk SSSR 233 (6), 1024–1027 (1977)].

    MATH  Google Scholar 

  32. A. M. Vershik and S. V. Kerov, “Characters and factor representations of the infinite symmetric group,” Sov. Math. Dokl. 23, 389–392 (1982) [transl. from Dokl. Akad. Nauk SSSR 257 (5), 1037–1040 (1981)].

    MATH  Google Scholar 

  33. A. M. Vershik and S. V. Kerov, “Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group,” Funct. Anal. Appl. 19(1), 21–31 (1985) [transl. from Funkts. Anal. Prilozh. 19 (1), 25–36 (1985)].

    Article  Google Scholar 

  34. A. M. Vershik and S. V. Kerov, “Locally semisimple algebras. Combinatorial theory and the K 0-functor,” J. Sov. Math. 38(2), 1701–1733 (1987) [transl. from Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Noveishie Dostizheniya 26, 3–56 (1985)].

    Article  Google Scholar 

  35. A. M. Vershik and S. V. Kerov, “The characters of the infinite symmetric group and probability properties of the Robinson–Schensted–Knuth algorithm,” SIAM J. Algebr. Discrete Methods 7, 116–124 (1986).

    Article  MathSciNet  Google Scholar 

  36. A. M. Vershik and A. A. Shmidt, “Limit measures arising in the asymptotic theory of symmetric groups. I,” Theory Probab. Appl. 22(1), 70–85 (1977) [transl. from Teor. Veroyatn. Primen. 22 (1), 72–88 (1977)].

    Article  Google Scholar 

  37. A. M. Vershik and A. A. Shmidt, “Limit measures arising in the asymptotic theory of symmetric groups. II,” Theory Probab. Appl. 23(1), 36–49 (1978) [transl. from Teor. Veroyatn. Primen. 23 (1), 42–54 (1978)].

    Article  Google Scholar 

  38. A. M. Whitney, “A reduction theorem for totally positive matrices,” J. Anal. Math. 2, 88–92 (1952).

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work is supported by the Russian Science Foundation under grant 17-71-20153 and performed at the St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, nab. Fontanki 27, St. Petersburg, Russia

    A. M. Vershik

  2. Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504, Russia

    A. M. Vershik

  3. Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, str. 1, Moscow, 127051, Russia

    A. M. Vershik

Authors
  1. A. M. Vershik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Vershik.

Additional information

To V. M. Buchstaber with friendly wishes on the occasion of his 75th birthday

This article was submitted by the author simultaneously in Russian and English

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 305, pp. 71–85.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vershik, A.M. Three Theorems on the Uniqueness of the Plancherel Measure from Different Viewpoints. Proc. Steklov Inst. Math. 305, 63–77 (2019). https://doi.org/10.1134/S0081543819030052

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543819030052

Search

Navigation