The isotropy group of the matrix multiplication tensor

TL;DR

This paper investigates the isotropy group of the matrix multiplication tensor, clarifying and formalizing previous results to facilitate the development of symmetry-aware fast matrix multiplication algorithms.

Contribution

It enlarges and refines the understanding of the isotropy group of the matrix multiplication tensor using group action language, providing a foundation for symmetry-based algorithm design.

Findings

Revised and formalized previous results on isotropy groups

Clarified the structure of the isotropy group for matrix multiplication tensors

Provided a rigorous framework for symmetry-based fast matrix multiplication algorithms

Abstract

By an {\em isotropy group} of a tensor $t\in V_1 \otimes V_2\otimes V_3=\widetilde V$ we mean the group of all invertible linear transformations of $\widetilde V$ that leave $t$ invariant and are compatible (in an obvious sense) with the structure of tensor product on~$\widetilde V$. We consider the case where $t$ is the structure tensor of multiplication map of rectangular matrices. The isotropy group of this tensor was studied in 1970s by de Groote, Strassen, and Brockett-Dobkin. In the present work we enlarge, make more precise, expose in the language of group actions on tensor spaces, and endow with proofs the results previously known. This is necessary for studying the algorithms of fast matrix multiplication admitting symmetries. The latter seems to be a promising new way for constructing fast algorithms.