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Abstract

This is the second in a series of papers on rank decompositions of the matrix
multiplication tensor. We present new rank $23$ decompositions for the $3\times
3$ matrix multiplication tensor $M_{\langle 3\rangle}$. All our decompositions
have symmetry groups that include the standard cyclic permutation of factors
but otherwise exhibit a range of behavior. One of them has 11 cubes as summands
and admits an unexpected symmetry group of order 12. We establish basic
information regarding symmetry groups of decompositions and outline two
approaches for finding new rank decompositions of $M_{\langle n\rangle}$ for
larger $n$.