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

TL;DR

This paper proves that no short, group-invariant algorithm with length ≤23 exists for multiplying 3x3 matrices, using orbit analysis of group actions on tensor decompositions.

Contribution

It establishes a lower bound on the multiplicative length of 3x3 matrix multiplication algorithms invariant under a specific group, extending previous orbit classification results.

Findings

No invariant algorithm of length ≤23 exists for 3x3 matrix multiplication.

Orbit analysis of group actions on tensor space was used in the proof.

Provides a group-theoretic lower bound for matrix multiplication complexity.

Abstract

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.