Rank and border rank of Kronecker powers of tensors and Strassens laser method

We prove that the border rank of the Kronecker square of the little Coppersmith-Winograd tensor $T_{cw,q}$ is the square of its border rank for $q > 2$ and that the border rank of its Kronecker cube is the cube of its border rank for $q > 4$. This answers questions raised implicitly in [Coppersmith-Winograd, 1990] and explicitly in [Bl"aser, 2013] and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith-Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen’s laser method, introducing a skew-symmetric version of the Coppersmith-Winograd tensor, $T_{skewcw,q}$. For $q = 2$, the Kronecker square of this tensor coincides with the $3\times 3$ determinant polynomial, $\det_3 \in \mathbb{C}^9\otimes \mathbb{C}^9\otimes \mathbb{C}^9$, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of $\det_3$, exhibiting a strict submultiplicative behaviour for $T_{skewcw,2}$ which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in $\mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3$.

💡 Research Summary

The paper investigates two central problems in the algebraic approach to matrix multiplication complexity: (1) the exact border rank of Kronecker powers of the small Coppersmith‑Winograd tensor (T_{cw,q}), and (2) the identification of new tensors that could be useful for Strassen’s laser method.

Background. For a tensor (T\in A\otimes B\otimes C) the rank (R(T)) is the minimal number of simple tensors needed to express (T), while the border rank (\underline R(T)) is the minimal number of simple tensors needed to approximate (T) arbitrarily closely. The Kronecker product (T\boxtimes T’) yields a new three‑way tensor, and the asymptotic rank (R^\infty(T)=\lim_{N\to\infty}R(T^{\boxtimes N})^{1/N}) controls the exponent (\omega) of matrix multiplication via (\omega=\log_2 R^\infty(M_{\langle n\rangle})). Strassen’s laser method uses an auxiliary tensor (T) with small rank and a large Kronecker power that degenerates to a matrix‑multiplication tensor; the method’s power hinges on sub‑multiplicativity of border rank under Kronecker powers and on the ability of (T^{\boxtimes N}) to degenerate to a large matrix‑multiplication tensor.

Main Result 1 – Exact border rank of (T_{cw,q}^{\boxtimes 2}) and (T_{cw,q}^{\boxtimes 3}). The authors prove that for all (q>2), \