Timeline for What is the minimum number of multiplications for $2\times 3$ and $3\times 2$ multiplication?

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when toggle format	what		by	license	comment
May 6, 2022 at 12:18	vote	accept	Turbo
May 6, 2022 at 7:06	answer	added	Zach Teitler		timeline score: 6
May 6, 2022 at 3:35	comment	added	Turbo		What is the representation for $k\times 2$ with $2\times 2$ and $2\times 2$ with $2\times k$?
May 6, 2022 at 3:22	comment	added	Zach Teitler		Isn't $M_{\langle a,b,c \rangle} = M_{\langle a,c,b \rangle} = M_{\langle b,c,a\rangle} = \dotsb$, it is invariant under all permutations of the quantities? See section 2.5.2 in Landsberg's book (first few chapters available from the author: math.tamu.edu/~joseph.landsberg/Tbookintro.pdf). For multiplication of $2 \times k$ times $k \times 2$, I think their notation is: $M_{\langle k,2,2 \rangle}$, and by symmetry that is the same as $M_{\langle 2,2,k \rangle}$. Does that sound right?
May 6, 2022 at 2:31	comment	added	Zach Teitler		In arxiv.org/abs/1911.07981, the tensor representing multiplication of $2 \times 3$ with $3 \times 2$ matrices is shown to have border rank $10$. They quote results of Landsberg-Ottaviani and Smirnov, that $2 \times k$ times $k \times 2$ matrix multiplication has border rank at least $3k$, and in some cases (conjecturally all cases) at most $3k+1$.
May 6, 2022 at 1:38	history	asked	Turbo	CC BY-SA 4.0