We discuss various notions of isomorphism for different kinds of tensors (such as plain, symmetric, alternating, etc.) as well as related problems such as algorithmically testing isomorphism of finite-dimensional algebras, groups, and spaces of matrices. We give polynomial-time algebraic reductions between all of these problems, showing that they are all equivalent. For example, we show how to convert d-tensors into 3-tensors such that two d-tensors are isomorphic if and only if the corresponding 3-tensors are isomorphic. Originally motivated by many problems in complexity theory, we hope that these transformations help others view their tensors from different angles and perhaps make connections between techniques for studying different "kinds" of tensors.
Joint work with V. Futorny and V. V. Sergeichuk (Lin. Alg. Appl. https://doi.org/10.1016/j.laa.2018.12.022) and Y. Qiao (https://arxiv.org/abs/1907.00309, one part in ITCS '21 and another to appear in CCC '21).
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