论文标题
差异组和张量类别
Oligomorphic groups and tensor categories
论文作者
论文摘要
给定差异群$ g $和$ g $的度量$ $ $（从某种意义上说），我们定义了一个刚性张量的类别$ \ upessline {\ mathrm {perm}}（g;μ）$的“置换模块”，并且在某些情况下，Abelian nline Line Line Line Line Line Line {此类别。当$ g $是无限的对称组时，这将恢复Deligne的插值类别。 $ g $的其他选择会导致新的张量类别。例如，我们以阳性特征构建了第一个已知的半简单前Tannakian类别，并具有超指定增长。我们结构的一个有趣方面是，与以前在这个方向上的工作不同，我们的类别是具体的：对象是环上的模块，张量产品会收到通用的双线性图。我们结构的核心是对纯态群体整合的新颖理论，这可能具有更大的兴趣。对纯态群体上的度量进行分类似乎是一个困难的问题，我们仅在少数情况下就解决了这一点。
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.
