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On subfactors arising from asymptotic representations of symmetric groups


Author: Makoto Yamashita
Journal: Proc. Amer. Math. Soc. 140 (2012), 249-261
MSC (2000): Primary 46L37; Secondary 20C32, 46L55
Published electronically: May 20, 2011
MathSciNet review: 2833537
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Abstract: We consider the infinite symmetric group and its infinite index subgroup given as the stabilizer subgroup of one element under the natural action on a countable set. This inclusion of discrete groups induces a hyperfinite subfactor for each finite factorial representation of the larger group. We compute subfactor invariants of this construction in terms of the Thoma parameter.


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  • [1] M. S. Boyko and N. I. Nessonov, Entropy of the shift on 𝐼𝐼₁-representations of the group 𝑆(∞), Ukr. Mat. Visn. 2 (2005), no. 1, 15–37; English transl., Ukr. Math. Bull. 2 (2005), no. 1, 15–37. MR 2172640
  • [2] Marek Bożejko and Mădălin Guţă, Functors of white noise associated to characters of the infinite symmetric group, Comm. Math. Phys. 229 (2002), no. 2, 209–227. MR 1923173, 10.1007/s00220-002-0687-2
  • [3] Rolf Gohm and Claus Köstler, Noncommutative independence from characters of the infinite symmetric group $ \mathbb{S}_\infty$, 2010. Preprint available at arXiv:1005.5726.
  • [4] Takeshi Hirai, Centralization of positive definite functions, weak containment of representations and Thoma characters for the infinite symmetric group, J. Math. Kyoto Univ. 44 (2004), no. 4, 685–713. MR 2118037
  • [5] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, 10.1007/BF01389127
  • [6] Sergei Kerov, Andrei Okounkov, and Grigori Olshanski, The boundary of the Young graph with Jack edge multiplicities, Internat. Math. Res. Notices 4 (1998), 173–199. MR 1609628, 10.1155/S1073792898000154
  • [7] Sergei Kerov, Grigori Olshanski, and Anatoly Vershik, Harmonic analysis on the infinite symmetric group, Invent. Math. 158 (2004), no. 3, 551–642. MR 2104794, 10.1007/s00222-004-0381-4
  • [8] Sergey Neshveyev and Erling Størmer, Dynamical entropy in operator algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 50, Springer-Verlag, Berlin, 2006. MR 2251116
  • [9] A. Yu. Okun′kov, Thoma’s theorem and representations of an infinite bisymmetric group, Funktsional. Anal. i Prilozhen. 28 (1994), no. 2, 31–40, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 28 (1994), no. 2, 100–107. MR 1283250, 10.1007/BF01076496
  • [10] A. Okun′kov, On representations 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, 166–228, 294 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 96 (1999), no. 5, 3550–3589. MR 1691646, 10.1007/BF02175834
  • [11] Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811
  • [12] Geoffrey Price, Extremal traces on some group-invariant 𝐶*-algebras, J. Funct. Anal. 49 (1982), no. 2, 145–151. MR 680655, 10.1016/0022-1236(82)90075-1
  • [13] Elmar Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 40–61 (German). MR 0173169
  • [14] 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
  • [15] A. M. Vershik and S. V. Kerov, Characters and factor representations of the infinite symmetric group, Dokl. Akad. Nauk SSSR 257 (1981), no. 5, 1037–1040 (Russian). MR 614033
  • [16] A. M. Vershik and S. V. Kerov, The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of 𝐾₀-functor of AF-algebras), Representation of Lie groups and related topics, Adv. Stud. Contemp. Math., vol. 7, Gordon and Breach, New York, 1990, pp. 39–117. MR 1104274
  • [17] Antony Wassermann, Automorphic actions of compact groups on operator algebras, Ph.D. thesis, 1981.

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Additional Information

Makoto Yamashita
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914, Japan
Email: makotoy@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10991-2
Keywords: Subfactor, symmetric group, asymptotic representation
Received by editor(s): November 27, 2009
Received by editor(s) in revised form: June 15, 2010, July 13, 2010, and November 10, 2010
Published electronically: May 20, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.