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