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.
Citation:
Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type”, Russian Math. Surveys, 70:4 (2015), 715–773
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\by Yu.~A.~Neretin
\paper Infinite symmetric groups and combinatorial constructions of topological field theory type
\jour Russian Math. Surveys
\yr 2015
\vol 70
\issue 4
\pages 715--773
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Linking options:
https://www.mathnet.ru/eng/rm9667
https://doi.org/10.1070/RM2015v070n04ABEH004958
https://www.mathnet.ru/eng/rm/v70/i4/p143
This publication is cited in the following 11 articles:
Yu. A. Neretin, “On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups”, Math. Notes, 114:4 (2023), 583–592
Neretin Yu.A., “On Spherical Unitary Representations of Groups of Spheromorphisms of Bruhat-Tits Trees”, Group. Geom. Dyn., 15:3 (2021), 801–824
Neretin Yu.A., “Groups Gl(Infinity) Over Finite Fields and Multiplications of Double Cosets”, J. Algebra, 585 (2021), 370–421
Neretin Yu.A., “Description of Unitary Representations of the Group of Infinite P-Adic Integer Matrices”, Represent. Theory, 25 (2021), 606–643
Yu. A. Neretin, “On the group of spheromorphisms of a homogeneous non-locally finite tree”, Izv. Math., 84:6 (2020), 1161–1191
A. A. Gaifullin, Yu. A. Neretin, “Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures”, J. Topol. Anal., 10:3 (2018), 605–625
Pablo Gonzalez Pagotto, “A Product on Double Cosets of B∞”, SIGMA, 14 (2018), 134, 18 pp.
J. Math. Sci. (N. Y.), 240:5 (2019), 572–586
Yu. A. Neretin, “Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument”, Funct. Anal. Appl., 51:2 (2017), 98–111
J. Math. Sci. (N. Y.), 232:2 (2018), 138–156
Yu. A. Neretin, “Several remarks on groups of automorphisms of free groups”, J. Math. Sci. (N. Y.), 215:6 (2016), 748–754
Citation in format AMSBIB
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