Geodesic
Non-existence of a short algorithm for multiplication of $3\times3$ matrices with group $S_4\times S_3$
Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 99-116

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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. 
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     title = {Non-existence of a short algorithm for multiplication of $3\times3$ matrices with group $S_4\times S_3$},
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V. P. Burichenko. Non-existence of a short algorithm for multiplication of $3\times3$ matrices with group $S_4\times S_3$. Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 99-116. http://geodesic.mathdoc.fr/item/TIMB_2022_30_1_a9/