Wildness for tensors

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 $[u_{i{i}^{\prime }}]$, $[v_{j{j}^{\prime }}]$, $[w_{k{k}^{\prime }}]$ are nonsingular matrices: this problem contains the problem of classifying an arbitrary system of tensors of order at most three.

arXiv comment38 pages

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Cited by in corpus (5)

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• Congruence of matrix spaces, matrix tuples, and multilinear maps
• General Linear Group Action on Tensors: A Candidate for Post-Quantum Cryptography
• Centers of Multilinear Forms and Applications