# A Finite-Tame-Wild Trichotomy Theorem for Tensor Diagrams

## TL;DR

This paper classifies tensor diagrams based on their complexity, showing that diagrams with vertices of degree three or more are wild, while others are finite or tame, aiding understanding of tensor network equivalences.

## Contribution

It establishes a trichotomy theorem for tensor diagrams, linking their structure to representation-theoretic complexity, and classifies indecomposables for finite and tame types.

## Key findings

- Tensor diagrams with vertices of degree ≥3 are wild.
- Diagrams without such vertices are finite or tame.
- Classification of indecomposable representations for finite and tame types.

## Abstract

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.