Oligomorphic groups and tensor categories
Abstract
Given an oligomorphic group G G and a measure $\mu $ μ for G G (in a sense that we introduce), we define a rigid tensor category underlinePerm(G; μ ) of “permutation modules,” and, in certain cases, an abelian envelope underlinetextupRep(G; μ ) of this category. When G G is the infinite symmetric group, this recovers Deligne’s interpolation category. Other choices for G 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.
Related
- Commutative exact algebras and modular tensor categories
- G-crossed braided tensor categories of twisted modules
- Lie algebras in symmetric tensor categories
- Derived Reconstruction of Semisimple Tensor and Module Categories
- The coanalyticity of the reduced K-theory on modules over free tensor algebras of dualizable t-categories
BibTeX
@article{harman2026oligomorphic,
title = {{Oligomorphic groups and tensor categories}},
author = {Nate Harman and Andrew Snowden},
journal = {Inventiones mathematicae},
year = {2026},
month = sep,
doi = {10.1007/s00222-026-01452-2},
url = {https://doi.org/10.1007/s00222-026-01452-2},
note = {CC BY},
}
RIS
TY - JOUR
TI - Oligomorphic groups and tensor categories
AU - Nate Harman
AU - Andrew Snowden
PY - 2026
DA - 2026/09/29
JO - Inventiones mathematicae
DO - 10.1007/s00222-026-01452-2
UR - https://doi.org/10.1007/s00222-026-01452-2
AB - Given an oligomorphic group G G and a measure $\mu $ μ for G G (in a sense that we introduce), we define a rigid tensor category underlinePerm(G; μ ) of “permutation modules,” and, in certain cases, an abelian envelope underlinetextupRep(G; μ ) of this category. When G G is the infinite symmetric group, this recovers Deligne’s interpolation category. Other choices for G 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.
SN - 0020-9910
SN - 1432-1297
ER -