arXiv subjects
Publications and source records attributed to Vladimir P. Burichenko.
It is proved that there is no an algorithm for multiplication of $3\times3$ matrices of multiplicative length $\leq23$ that is invariant under a certain group isomorphic to $S_4\times S_3$. The proof makes use of description of the orbits of this group on decomposable tensors in the tensor cube $(M_3({\mathbb C}))^{\otimes3}$ which was obtained earlier.
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.
We consider the famous Strassen algorithm for fast multiplication of matrices. We show that this algorithm has a nontrivial finite group of automorphisms of order 36 (namely the direct product of two copies of the symmetric group on 3 symbols), or even 72, if we consider "extended" Strassen algorithm. This is an indirect evidence that the (unknown at present) optimal algorithm for multiplication of two size 3 by 3 matrices also may have a large automorphism group, and this may be a fruitful idea for a search of such an algorithm. In the beginning we give a brief introduction to the subject, to make the text accessible for specialists in the representation theory of finite groups.