Non-existence of a short algorithm for multiplication of $3\times3$ matrices with group $S_4\times S_3$

TL;DR

This paper proves that there are no matrix multiplication algorithms of length 23 or less for 3x3 matrices that admit the symmetry group S4×S3, by analyzing group orbits on tensor decompositions.

Contribution

It provides a complete classification of group orbits on tensors relevant to symmetric algorithms and demonstrates the non-existence of short algorithms with the specified symmetry.

Findings

No algorithms of length ≤23 exist with symmetry group S4×S3.

Complete orbit classification for tensors under the group action.

Methodology for ruling out symmetric algorithms of certain lengths.

Abstract

One of prospective ways to find new fast algorithms of matrix multiplication is to study algorithms admitting nontrivial symmetries. In the work possible algorithms for multiplication of $3\times3$ matrices, admitting a certain group $G$ isomorphic to $S_4\times S_3$, are investigated. It is shown that there exist no such algorithms of length $\leq23$. In the first part of the work, which is the content of the present article, we describe all orbits of length $\leq23$ of $G$ on the set of decomposable tensors in the space $M\otimes M\otimes M$, where $M=M_3({\mathbb C})$ is the space of complex $3\times3$ matrices. In the second part of the work this description will be used to prove that a short algorithm with the above-mentioned group does not exist.