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$S_{\infty }$ representations and combinatorial identities


Author: Amitai Regev
Journal: Trans. Amer. Math. Soc. 353 (2001), 4371-4404
MSC (2000): Primary 20C32
DOI: https://doi.org/10.1090/S0002-9947-01-02772-6
Published electronically: June 14, 2001
MathSciNet review: 1851175
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Abstract: For various probability measures on the space of the infinite standard Young tableaux we study the probability that in a random tableau, the $(i,j)^{th}$ entry equals a given number $n$. Beside the combinatorics of finite standard tableaux, the main tools here are from the Vershik-Kerov character theory of $S_{\infty}$. The analysis of these probabilities leads to many explicit combinatorial identities, some of which are related to hypergeometric series.


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  • [BR] A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. Math. 64 (1987), 118-175. MR 88i:20006
  • [B] A.M. Borodin, Multiplicative central measure on Schur graph, Zap. Nauc. Sem. POMI 240 (199), 44-52 (Russian). English translation in J. Math. Sci. 96 (1999), 3472-3477. MR 2000g:43001
  • [HH] P. Hoffman and J.F. Humphreys, Projective Representation of the Symmetric Groups, Oxford Mathematical Monographs, 1992. MR 94f:20047
  • [I] V.N Ivanov, Dimensions of skew shifted Young diagrams and projective characters of the infinite symmetric group, Zap. Nauc. Sem. POMI 240 (1997), 116-136 (Russian). English translation in J. Math. Sci. 96 (1999), 3517-3530. MR 2000j:05124
  • [J] G.D. James, The Representation Theory of the Symmetric Groups, Springer Lecture Notes in Mathematics 682 (1978). MR 80g:20019
  • [K] S. Kerov, Anisotropic Young diagrams and Jack symmetric functions, Funct. Anal. Appl. 34 (2000), 41-51. CMP 2000:12
  • [KOV] S. Kerov, G. Olshanski and A. Vershik, Harmonic analysis on the infinite symmetric group. A deformation of the regular representation, C.R. Acad. Sci. Paris, Ser. I, Math. 316 (8) (1993), 773-778. MR 94e:20019
  • [M] I.G. Macdonald, Symmetric Functions and Hall Polynomials, $2^{nd}$ Edition, Oxford, 1995. MR 96h:05207
  • [M1] I.G. Macdonald, Private letter.
  • [N] M. Nazarov, Projective representation of the infinite symmetric group, Representation Theory and Dynamical Systems (A. M. Vershik, editor), Advances in Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 115-130. MR 93j:20032
  • [O] G. Olshanski, Point processes and the infinite symmetric group. Part I: The general formalism and the density function, Preprint, math.RT/9804086, 1998. To appear.
  • [R] A. Regev, Asymptotics of degrees of some $S_{n}$-subregular representations, Isr. J. Math. 113 (1999), 15-28. MR 2001b:05226
  • [S] I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare substitutionen, J. Reine Angew. Math. 139 (1911), 155-250.
  • [T] E. Thoma, Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Zeitschr. 85 (1964), 40-61. MR 30:3382
  • [VK1] A.M. Vershik and S.V. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Dokl. Akad. Nauk SSSR 233 (1977), 1024-1027; English transl., Soviet Math. Dokl. 18 (1977), 527-531. MR 58:562
  • [VK2] A.M. Vershik and S. Kerov, Asymptotics of maximal and typical dimensions of irreducible representations of the symmetric group, Funkts. Anal. Prilozhen 19 (1985), 25-36; English transl., Funct. Anal. Appl. 19 (1985), 21-31. MR 86k:11051
  • [VK3] A.M Vershik and S. Kerov, Asymptotic theory of characters of the symmetric group, Funct. Anal. Appl. 15 (1981), 246-255. MR 84a:22016
  • [Z] D. Zeilberger, Shalosh, Private communication.

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Additional Information

Amitai Regev
Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel - and - Department of Mathematics, Pennsylvania State University, State College, Pennsylvania 16802
Address at time of publication: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
Email: regev@wisdom.weizmann.ac.il

DOI: https://doi.org/10.1090/S0002-9947-01-02772-6
Received by editor(s): March 14, 1999
Published electronically: June 14, 2001
Additional Notes: This work was partially supported by ISF grant 6629/1
Article copyright: © Copyright 2001 American Mathematical Society

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