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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 4(424), Pages 143–204 (Mi umn9667)  

This article is cited in 6 scientific papers (total in 6 papers)

Infinite symmetric groups and combinatorial constructions of topological field theory type

Yu. A. Neretinabcd

a University of Vienna, Vienna, Austria
b Institute for Theoretical and Experimental Physics, Moscow
c Moscow State University
d Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: This paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group $G$, a subgroup $K$) categories are constructed whose morphisms are two-dimensional surfaces tiled by polygons and coloured in a certain way. A product of morphisms is a gluing together of combinatorial bordisms, and functors from the category of bordisms to the category of Hilbert spaces and bounded operators correspond to unitary representations of $G$. The construction has numerous variations: instead of surfaces there can also be one-dimensional objects of Brauer diagram type, multidimensional pseudomanifolds, and bipartite graphs.
Bibliography: 66 titles.

Keywords: infinite symmetric group, representations of categories, spherical representations, double cosets, bordisms.

Funding Agency Grant Number
Austrian Science Fund P25142
This work was supported by the Austrian Science Fund FWF (grant no. P25142).


DOI: https://doi.org/10.4213/rm9667

Full text: PDF file (1258 kB)
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English version:
Russian Mathematical Surveys, 2015, 70:4, 715–773

Bibliographic databases:

UDC: 517.986.4+519.12+512.583
MSC: 20B30, 20C32
Received: 01.12.2014

Citation: Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type”, Uspekhi Mat. Nauk, 70:4(424) (2015), 143–204; Russian Math. Surveys, 70:4 (2015), 715–773

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. A. Neretin, “Several remarks on groups of automorphisms of free groups”, J. Math. Sci. (N. Y.), 215:6 (2016), 748–754  mathnet  crossref  mathscinet
    2. Yu. A. Neretin, “Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument”, Funct. Anal. Appl., 51:2 (2017), 98–111  mathnet  crossref  crossref  isi  elib
    3. J. Math. Sci. (N. Y.), 232:2 (2018), 138–156  mathnet  crossref
    4. A. A. Gaifullin, Yu. A. Neretin, “Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures”, J. Topol. Anal., 10:3 (2018), 605–625  crossref  mathscinet  isi  scopus
    5. P. Gonzalez Pagotto, “A product of double cosets of $B_\infty$”, SIGMA, 14 (2018), 134, 18 pp.  mathnet  crossref
    6. Yu. A. Neretin, “On the group of infinite $p$-adic matrices with integer elements”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXIX, Zap. nauchn. sem. POMI, 468, POMI, SPb., 2018, 105–125  mathnet
  • Успехи математических наук Russian Mathematical Surveys
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