Infinite symmetric groups and combinatorial constructions of topological field theory type

Yu. A. Neretin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 70 (2015) no. 4, pp. 715-773

Résumé

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.

Détail

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

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