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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 6, Pages 93–124 (Mi izv8410)  

This article is cited in 1 scientific paper (total in 1 paper)

Stable representations of the infinite symmetric group

A. M. Vershikabc, N. I. Nessonovd

a Saint Petersburg State University
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
d B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov

Abstract: We study the notion of a stable unitary representation of a group (or a $\star$-representation of a $\mathbf C^\star$-algebra) with respect to some group of automorphisms of the group (or algebra). In the case of the group of finitary permutations of a countable set we give a complete description, up to quasi-equivalence, of the representations which are stable with respect to the group of all automorphisms of the group. In particular, we solve an old question concerning factor representations associated with Ol'shansky–Okun'kov admissible representations. It is proved that these representations are induced by factor representations of type $II_1$ of two-block Young subgroups. The class of stable representations will be the subject of further research.

Keywords: infinite symmetric group, stable representations, factor representations, characters, semidirect product, groupoid model.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00373
13-01-12422-офи_м
The research of the first author was partially supported by the Russian Foundation for Basic Research (grants nos.~14-01-00373 and~13-01-12422-ofi-i), and that of the second author was supported by the grant `Network of Mathematical Research 2013--2015'.


DOI: https://doi.org/10.4213/im8410

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English version:
Izvestiya: Mathematics, 2015, 79:6, 1184–1214

Bibliographic databases:

UDC: 519.12+512.58
MSC: Primary 20C32; Secondary 20B30, 22A25
Received: 15.05.2015
Revised: 09.06.2015

Citation: A. M. Vershik, N. I. Nessonov, “Stable representations of the infinite symmetric group”, Izv. RAN. Ser. Mat., 79:6 (2015), 93–124; Izv. Math., 79:6 (2015), 1184–1214

Citation in format AMSBIB
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\paper Stable representations of the infinite symmetric group
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\pages 93--124
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\vol 79
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\pages 1184--1214
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    This publication is cited in the following articles:
    1. N. I. Nessonov, “An analogue of Schur–Weyl duality for the unitary group of a $\mathrm{II}_1$-factor”, Sb. Math., 210:3 (2019), 447–472  mathnet  crossref  crossref  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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