Submission date: 9 April 2022

Abstract:

Given an oligomorphic group G and a measure μ for G (in a sense that we introduce), we define a rigid tensor category Perm(G; &mu); of “permutation modules”, and, in certain cases, an abelian envelope Rep(G; &mu); of this category. When G is the infinite symmetric group, this recovers Deligne's interpolation category, while other choices for G lead to fundamentally new tensor categories. In particular, we give the first example (so far as we know) of a C-linear rigid tensor category that bears no obvious relation to finite dimensional representations. One interesting aspect of our construction is that, unlike much previous work in this direction, our categories are concrete: the objects are modules over a ring, and the tensor product receives a universal bi-linear map. Central to our constructions is a novel theory of integration on oligomorphic groups, which could be of more general interest. Classifying the measures on an oligomorphic group appears to be a difficult problem, which we solve in only a few cases.

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Full text arXiv 2204.04526: pdf, ps.