Oligomorphic groups and tensor categories

Given an oligomorphic group \(G\) and a measure \(\mu\) for \(G\) (in a sense that we introduce), we define a rigid tensor category \(\underline{\mathrm{Perm}}(G; \mu)\) of "permutation modules," and, in certain cases, an abelian envelope \(\underline{\mathrm{Rep}}(G; \mu)\) of this catego...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Harman, Nate, Snowden, Andrew
Format: Article
Language:English
Subjects:
Categories
Group theory
Interpolation
Mathematical analysis
Modules
Permutations
Tensors
Online Access:Get full text
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Summary:Given an oligomorphic group \(G\) and a measure \(\mu\) for \(G\) (in a sense that we introduce), we define a rigid tensor category \(\underline{\mathrm{Perm}}(G; \mu)\) of "permutation modules," and, in certain cases, an abelian envelope \(\underline{\mathrm{Rep}}(G; \mu)\) of this category. When \(G\) is the infinite symmetric group, this recovers Deligne's interpolation category. Other choices for \(G\) lead to fundamentally new tensor categories. For example, we construct the first known semi-simple pre-Tannakian categories in positive characteristic with super-exponential growth. One interesting aspect of our construction is that, unlike 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.
ISSN:2331-8422