In this paper, we consider the problem of determining when two tensor networks are equivalent under a heterogeneous change of basis. In particular, to a string diagram in a certain monoidal category (which we call tensor diagrams), we formulate an associated abelian category of representations. Each representation corresponds to a tensor network on that diagram. We then classify which tensor diagrams give rise to categories that are finite, tame, or wild in the traditional sense of representation theory. For those tensor diagrams of finite and tame type, we classify the indecomposable representations. Our main result is that a tensor diagram is wild if and only if it contains a vertex of degree at least three. Otherwise, it is of tame or finite type.

中文翻译：

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张量图的有限驯服三分定理

在本文中，我们考虑在异构基变化下确定两个张量网络何时等效的问题。特别是，对于某个幺半群范畴中的弦图（我们称之为张量图），我们制定了一个相关的阿贝尔表示范畴。每个表示对应于该图上的一个张量网络。然后，我们对哪些张量图产生了传统意义上的表示理论中的有限、驯服或狂野的类别进行分类。对于那些有限和驯服类型的张量图，我们对不可分解的表示进行分类。我们的主要结果是，当且仅当它包含度数至少为 3 的顶点时，张量图才是狂野的。否则，它是驯服或有限类型的。