Nonexistence of a short algorithm for multiplication of $3\times3$ matrices with group $S_4\times S_3$
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 abovementioned group does not exist.
 Publication:

arXiv eprints
 Pub Date:
 November 2022
 DOI:
 10.48550/arXiv.2211.03404
 arXiv:
 arXiv:2211.03404
 Bibcode:
 2022arXiv221103404B
 Keywords:

 Computer Science  Computational Complexity;
 68Q25;
 20C;
 F.2
 EPrint:
 19 pp. Accepted for publication in Proceedings of the Institute of mathematics (of Academy of Sciences of Belarus)