Wildness for tensors

TL;DR

This paper establishes that classifying three-dimensional tensors under equivalence transformations is a wild problem, meaning it encompasses the complexity of classifying arbitrary systems of tensors and is considered hopeless.

Contribution

It proves that the tensor classification problem of order at most three is universal and as complex as the wild problems in representation theory.

Findings

Tensor classification problem is wild for order at most three.

Classifying three-dimensional arrays encompasses all tensor classification problems.

The problem contains the classification of arbitrary systems of tensors of order up to three.

Abstract

In representation theory, a classification problem is called wild if it contains the problem of classifying matrix pairs up to simultaneous similarity. The latter problem is considered as hopeless; it contains the problem of classifying an arbitrary finite system of vector spaces and linear mappings between them. We prove that an analogous "universal" problem in the theory of tensors of order at most 3 over an arbitrary field is the problem of classifying three-dimensional arrays up to equivalence transformations \[ [a_{ijk}]_{i=1}^{m}\,{}_{j=1}^{n}\,{}_{k=1}^{t}\ \mapsto\ \Bigl[ \sum_{i,j,k} a_{ijk}u_{ii'} v_{jj'}w_{kk'}\Bigr]{}_{i'=1}^{m}\,{}_{j'=1}^{n}\,{}_{k'=1}^{t} \] in which $[u_{ii'}]$, $[v_{jj'}]$, $[w_{kk'}]$ are nonsingular matrices: this problem contains the problem of classifying an arbitrary system of tensors of order at most three.