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Four drafts on the representation theory of the group of infinite matrices over a finite field

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Abstract

The history of these drafts is described in the preface written by the first author; the drafts were written in 1997–2000, when the authors studied asymptotic representation theory. Each draft is devoted to one of the main chapters of the representation theory of the group of infinite matrices over a finite field. The list of results includes the definition of the group GLB(q), which is the right analog of the group of matrices over a finite field, or a q-analog of the infinite symmetric group, the latter corresponding to q = 1; a formula for the principal series characters of the group GLB(q) and similar groups; the statistics of Jordan forms and a law of large numbers for the characters; a construction of simplest factor representations. Bibliography: 21 titles.

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References

  1. A. M. Borodin, “Law of large numbers and central limit theorem for Jordan normal form of large triangular matrices over a finite field,” J. Math. Sci. (New York), 96,No. 5, 3455–3471 (1999).

    Article  MathSciNet  Google Scholar 

  2. D. K. Faddeev, “Complex representations of the general linear group over a finite field,” J. Sov. Math., 9,No. 3, 64–78 (1978).

    MathSciNet  Google Scholar 

  3. J. A. Green, “The characters of the finite general linear groups,” Trans. Amer. Math. Soc., 80, 402–447 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  4. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis on Groups, Vol. II, Springer-Verlag, New York-Berlin (1970).

    MATH  Google Scholar 

  5. S. V. Kerov, “Generalized Hall-Littlewood symmetric functions and orthogonal polynomials,” Adv. Soviet Math., 9, 67–94 (1992).

    MathSciNet  Google Scholar 

  6. S. V. Kerov, Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis, Amer. Math. Soc., Providence, Rhode Island (2003).

    MATH  Google Scholar 

  7. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford (1979).

    MATH  Google Scholar 

  8. J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton (1974).

    MATH  Google Scholar 

  9. J. Renault, A Groupoid Approach to C*-Algebras, Lect. Notes Math., 793, Springer, Berlin (1980).

    MATH  Google Scholar 

  10. H. L. Skudlarek, “Die unzerlegbaren Charactere einiger diskreter Gruppen,” Math. Ann., 223, 213–231 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  11. R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1986).

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. E. Thoma, “Die Einschränkung der Charactere von GL(n, q) auf GL(n−1, q),” Math. Z., 119, 321–338 (1971).

    Article  MathSciNet  Google Scholar 

  14. A. M. Vershik, “Theory of decreasing sequences of measurable partitions,” St. Petersburg Math. J., 6,No. 4, 705–761 (1995).

    MathSciNet  Google Scholar 

  15. A. M. Vershik, “Two lectures on the asymptotic representation theory and statistics of Young diagrams,” Lect. Notes Math., 1815, 161–182 (2003).

    Article  MathSciNet  Google Scholar 

  16. A. M. Vershik and S. V. Kerov, “Characters and factor representations of the infinite symmetric group,” Sov. Math. Dokl., 23, 389–392 (1981).

    MATH  Google Scholar 

  17. A. M. Vershik and S. V. Kerov, “Characters and factor representations of the infinite unitary group,” Sov. Math. Dokl., 26, 570–574 (1982).

    MATH  Google Scholar 

  18. A. M. Vershik and S. V. Kerov, “Asymptotic theory of characters of the symmetric group,” Funct. Anal. Appl., 15, 246–255 (1982).

    Article  Google Scholar 

  19. A. M. Vershik and S. V. Kerov, “Locally semisimple algebras. Combinatorial theory and the K 0-functor,” in: Itogi Nauki i Tekhniki, Ser. Sovrem. Probl. Mat., 26, VINITI, Moscow (1985), pp. 3–56. English translation: J. Sov. Math., 38, 1701–1733 (1987).

    Google Scholar 

  20. A. M. Vershik and S. V. Kerov, “On an infinite-dimensional group over a finite field,” Funct. Anal. Appl., 32,No. 3, 147–152 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  21. A. V. Zelevinsky, Representations of Finite Classical Groups. A Hopf algebra approach, Lect. Notes Math., 869, Springer-Verlag, Berlin-New York (1981).

    MATH  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia

    A. M. Vershik & S. V. Kerov

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  1. A. M. Vershik
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  2. S. V. Kerov
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Correspondence to A. M. Vershik.

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 5–36.

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Vershik, A.M., Kerov, S.V. Four drafts on the representation theory of the group of infinite matrices over a finite field. J Math Sci 147, 7129–7144 (2007). https://doi.org/10.1007/s10958-007-0535-1

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  • DOI: https://doi.org/10.1007/s10958-007-0535-1

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