Wildness for tensors

Author(s)

Futorny, Vyacheslav ^[1] ; Grochow, Joshua A. ^{[2, 3]} ; Sergeichuk, Vladimir V. ^[4]

Affiliation

^[1] Univ Sao Paulo, Dept Math, Sao Paulo - Brazil

^[2] Univ Colorado, Dept Comp Sci, Boulder, CO 80309 - USA

^[3] Univ Colorado, Dept Math, Boulder, CO 80309 - USA

^[4] Inst Math, Kiev - Ukraine

• Document type: Journal article
• Source: Linear Algebra and its Applications; v. 566, p. 212-244, APR 1 2019.
• Web of Science Citations: 1

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 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) (n)(j=1) (t)(k=1) bar right arrow {[}Sigma(i,j,k) a(ijk) u(ii') v(jj') w(kk')](i'=1)(m) (n)(j'=1) (t)(k'=1) 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. (C) 2018 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 14/09310-5 - Algebraic structures and their representations
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 15/05864-9 - Classification problems in linear algebra and system theory
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Grants - Visiting Researcher Grant - International