$S_{\infty }$ representations and combinatorial identities
HTML articles powered by AMS MathViewer
- by Amitai Regev
- Trans. Amer. Math. Soc. 353 (2001), 4371-4404
- DOI: https://doi.org/10.1090/S0002-9947-01-02772-6
- Published electronically: June 14, 2001
- PDF | Request permission
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.References
- A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118–175. MR 884183, DOI 10.1016/0001-8708(87)90007-7
- A. M. Borodin, Multiplicative central measures on the Schur graph, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240 (1997), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, 44–52, 290–291 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 96 (1999), no. 5, 3472–3477. MR 1691636, DOI 10.1007/BF02175824
- A. M. Brunner, R. G. Burns, and Sheila Oates-Williams, On almost primitive elements of free groups with an application to Fuchsian groups, Canad. J. Math. 45 (1993), no. 2, 225–254. MR 1208114, DOI 10.4153/CJM-1993-011-9
- V. N. Ivanov, The dimension of skew shifted Young diagrams, and projective characters of the infinite symmetric group, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240 (1997), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, 115–135, 292–293 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 96 (1999), no. 5, 3517–3530. MR 1691642, DOI 10.1007/BF02175830
- G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
- S. Kerov, Anisotropic Young diagrams and Jack symmetric functions, Funct. Anal. Appl. 34 (2000), 41–51.
- Serguei Kerov, Grigori Olshanski, and Anatoli Vershik, Harmonic analysis on the infinite symmetric group. A deformation of the regular representation, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 8, 773–778 (English, with English and French summaries). MR 1218259
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- I.G. Macdonald, Private letter.
- M. L. Nazarov, Projective representations of the infinite symmetric group, Representation theory and dynamical systems, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 115–130. MR 1166198
- 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.
- Amitai Regev, Asymptotics of degrees of some $S_n$-sub regular representations, Israel J. Math. 113 (1999), 15–28. MR 1729440, DOI 10.1007/BF02780170
- I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare substitutionen, J. Reine Angew. Math. 139 (1911), 155–250.
- Elmar Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 40–61 (German). MR 173169, DOI 10.1007/BF01114877
- A. M. Veršik and S. V. Kerov, Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Dokl. Akad. Nauk SSSR 233 (1977), no. 6, 1024–1027 (Russian). MR 0480398
- A. M. Vershik and S. V. Kerov, Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25–36, 96 (Russian). MR 783703
- A. M. Vershik and S. V. Kerov, Asymptotic theory of the characters of a symmetric group, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 15–27, 96 (Russian). MR 639197
- D. Zeilberger, Shalosh, Private communication.
Bibliographic 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
- Received by editor(s): March 14, 1999
- Published electronically: June 14, 2001
- Additional Notes: This work was partially supported by ISF grant 6629/1
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 4371-4404
- MSC (2000): Primary 20C32
- DOI: https://doi.org/10.1090/S0002-9947-01-02772-6
- MathSciNet review: 1851175