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 $\mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3$.

翻译：我们证明小铜匠{维诺格 {维诺格 $T ⁇ cw,q} 美元的Kronecker广场的边界位置是其边界位置的正方位,$ > 2美元, Kronecker 立方的边界位置是其边界位置的立方, $ > 4美元。 这个答案隐含地在[ Coppersmith- Winograd, 1990] 和[ Bl\'aser, 2013] 中提出。 排除了使用小铜匠- 维诺格 的平方或立方的基块来证明矩阵增殖的新界限的可能性。 在正正向方向, 我们扩大清晰的高压列表列表列表, 可能用于 Strassen的激光法, $\\ 维诺格格阵列的立方- 数 。 用于显示 共振- 共振- 立方格 的共立方( 共立方 3 美元) 的直立方位 方格 方位 和共立体 立体 立体的直立体 立体 立体 。