A Finite-Tame-Wild Trichotomy Theorem for Tensor Diagrams

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.