On automorphism group of a possible short algorithm for multiplication of $3\times3$ matrices

TL;DR

This paper investigates the symmetry properties of potential short algorithms for multiplying 3x3 matrices, establishing a bound on their automorphism groups if such algorithms exist with length up to 22.

Contribution

It proves that any such algorithm's automorphism group must be a subgroup of a specific symmetric group product, constraining the structure of possible algorithms.

Findings

Automorphism group is a subgroup of S_l x S_3 for algorithms with length ≤ 22.

Provides structural constraints on symmetry groups of short matrix multiplication algorithms.

Advances understanding of symmetry in matrix multiplication algorithms.

Abstract

Studying algorithms admitting nontrivial symmetries is a prospective way of constructing new short algorithms of matrix multiplication. The main result of the article is that if there exists an algorithm of multiplicative length $l\leq22$ for multuplication of $3\times3$ matrices then its automorphism group is isomorphic to a subgroup of $S_l\times S_3$.