New lower bounds for matrix multiplication and the 3x3 determinant

Abstract

Let $M_{\langle u,v,w\rangle}\in C^{uv}\otimes C^{vw}\otimes C^{wu}$ denote the matrix multiplication tensor (and write $M_n=M_{\langle n,n,n\rangle}$) and let $det_3\in ( C^9)^{\otimes 3}$ denote the determinant polynomial considered as a tensor. For a tensor $T$, let $\underline R(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $\underline R(M_2)=7$,(ii) prove $\underline R(M_{\langle 223\rangle})=10$, and $\underline R(M_{\langle 233\rangle})=14$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_2$,(iii) prove $\underline R( M_3)\geq 17$, (iv) prove $\underline R( det_3)=17$, improving the previous lower bound of $12$, (v) prove $\underline R(M_{\langle 2nn\rangle})\geq n^2+1.32n$ for all $n\geq 25$ (previously only $\underline R(M_{\langle 2nn\rangle})\geq n^2+1$ was known) as well as lower bounds for $4\leq n\leq 25$, and (vi) prove $\underline R(M_{\langle 3nn\rangle})\geq n^2+2 n+1$ for all $ n\geq 21$, where previously only $\underline R(M_{\langle 3nn\rangle})\geq n^2+2$ was known, as well as lower boundsfor $4\leq n\leq 21$. Our results utilize a new technique initiated by Buczy\'{n}ska and Buczy\'{n}ski, called border apolarity. The two key ingredients are: (i) the use of a multi-graded ideal associated to a border rank $r$ decomposition of any tensor, and (ii) the exploitation of the large symmetry group of $T$ to restrict to $B_T$-invariant ideals, where $B_T$ is a maximal solvable subgroup of the symmetry group of $T$.