Oligomorphic groups and tensor categories
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909156298981376 |
|---|---|
| author | Harman, Nate Snowden, Andrew |
| author_facet | Harman, Nate Snowden, Andrew |
| contents | Given an oligomorphic group $G$ and a measure $μ$ for $G$ (in a sense that we introduce), we define a rigid tensor category $\underline{\mathrm{Perm}}(G; μ)$ of "permutation modules," and, in certain cases, an abelian envelope $\underline{\mathrm{Rep}}(G; μ)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_04526 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Oligomorphic groups and tensor categories Harman, Nate Snowden, Andrew Representation Theory Combinatorics Logic Given an oligomorphic group $G$ and a measure $μ$ for $G$ (in a sense that we introduce), we define a rigid tensor category $\underline{\mathrm{Perm}}(G; μ)$ of "permutation modules," and, in certain cases, an abelian envelope $\underline{\mathrm{Rep}}(G; μ)$ 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. |
| title | Oligomorphic groups and tensor categories |
| topic | Representation Theory Combinatorics Logic |
| url | https://arxiv.org/abs/2204.04526 |