@online{Ballard_arXiv1801.00843,
TITLE = {{The Geometry of Rank Decompositions of Matrix Multiplication II: $3\times 3$ Matrices}},
AUTHOR = {Ballard, Grey and Ikenmeyer, Christian and Landsberg, J. M. and Ryder, Nick},
LANGUAGE = {eng},
URL = {http://arxiv.org/abs/1801.00843},
EPRINT = {1801.00843},
EPRINTTYPE = {arXiv},
YEAR = {2018},
ABSTRACT = {This is the second in a series of papers on rank decompositions of the matrix<br>multiplication tensor. We present new rank $23$ decompositions for the $3\times<br>3$ matrix multiplication tensor $M_{\langle 3\rangle}$. All our decompositions<br>have symmetry groups that include the standard cyclic permutation of factors<br>but otherwise exhibit a range of behavior. One of them has 11 cubes as summands<br>and admits an unexpected symmetry group of order 12. We establish basic<br>information regarding symmetry groups of decompositions and outline two<br>approaches for finding new rank decompositions of $M_{\langle n\rangle}$ for<br>larger $n$.<br>},
}