New lower bounds for matrix multiplication and the 3x3 determinant

Abstract

Let M ⟨u,v,w⟩ ∈ C uv ⊗C vw ⊗C wu denote the matrix multiplication tensor (and write M ⟨n⟩ = M ⟨n,n,n⟩ ) and let det3 ∈ (C 9 ) ⊗3 denote the determinant polynomial considered as a tensor. For a tensor T , let R(T ) denote its border rank. We (i) give the first hand-checkable algebraic proof that R(M ⟨2⟩ ) = 7, (ii) prove R(M ⟨223⟩ ) = 10, and R(M ⟨233⟩ ) = 14, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was M ⟨2⟩ , (iii) prove R(M ⟨3⟩ ) ≥ 17, (iv) prove R(det3) = 17, improving the previous lower bound of 12, (v) prove R(M ⟨2nn⟩ ) ≥ n 2 + 1.32n for all n ≥ 25 (previously only R(M ⟨2nn⟩ ) ≥ n 2 + 1 was known) as well as lower bounds for 4 ≤ n ≤ 25, and (vi) prove R(M ⟨3nn⟩ ) ≥ n 2 + 2n for all n ≥ 21, where previously only R(M ⟨3nn⟩ ) ≥ n 2 + 2 was known, as well as lower bounds for 4 ≤ n ≤ 21,.Our results utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to tensors with symmetry to obtain an algorithm that, given a tensor T with a large symmetry group and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all potential border rank r decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. 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 .

Two classes of minimal generic fundamental invariants for tensors

“…In Corollary 4.8, if we let = m = n = 3 and = 2, m = 3, n = 3, we have that F 3 ( 2, 3, 3 ) = 0. By Theorem 1.4 of [17], we know that the border rank of 2, 3, 3 is 14. On the other hand, the chromatic index of H = B(3, 3, 3) is 9, so in some sense Proposition 4.6 is better than Proposition 4.2 of [14].…”

On the structure tensor of $\mathfrak{sl}_n$

“…Border Apolarity. In order to establish larger lower bounds on R(T sl 3 ) than can be achieved by Koszul flattenings and border substitution for T sl 3 , we will use border apolarity, as developed in [BB20] and [CHL19].…”

“…In [CHL19], using Weak Border Apolarity Theorem, they assert that for T a concise tensor with a border rank r decomposition, there will exist an ideal I satisfying the following:…”

“…We note that the last condition is simply the condition that I is an ideal where the multiplication respects the grading. The border apolarity algorithm presented in [CHL19] makes use of these conditions and attempts to iteratively construct all possible ideals, I, in each multidegree. In particular, if at any multidegree ijk, there does not exist an I ijk satisfying the above, then we may conclude that R(T ) > r. We remark that the candidate ideals, I, do not necessarily correspond to an actual border rank decomposition of the tensor.…”

“…In particular, if at any multidegree ijk, there does not exist an I ijk satisfying the above, then we may conclude that R(T ) > r. We remark that the candidate ideals, I, do not necessarily correspond to an actual border rank decomposition of the tensor. We describe the algorithm of [CHL19] precisely below:…”