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The Geometry of Rank Decompositions of Matrix Multiplication II: 3 x 3 matrices

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons202366

Ikenmeyer,  Christian
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Fulltext (public)

arXiv:1801.00843.pdf
(Preprint), 319KB

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Citation

Ballard, G., Ikenmeyer, C., Landsberg, J. M., & Ryder, N. (2018). The Geometry of Rank Decompositions of Matrix Multiplication II: 3 x 3 matrices. Retrieved from http://arxiv.org/abs/1801.00843.


Cite as: http://hdl.handle.net/21.11116/0000-0001-3F64-9
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 $n$.