Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories

TL;DR

This paper explores the representation theory of the infinite tri-symmetric group, demonstrating how certain spherical representations generate functors from a category of surfaces to Hilbert spaces, linking algebraic and topological structures.

Contribution

It introduces a novel connection between infinite symmetric groups, surface categories, and topological field theories, expanding the understanding of their interplay.

Findings

Representation of infinite tri-symmetric group generates functors to Hilbert spaces.

Establishes a link between double coset multiplication and topological field theories.

Provides a new framework for studying algebraic structures via topological methods.

Abstract

We consider a product of three copies of infinite symmetric group and its representations spherical with respect to the diagonal subgroup. We show that such representations generate functors from a certain category of simplicial two-dimensional surfaces to the category of Hilbert spaces and bounded linear operators.