Separable symmetric tensors and separable anti-symmetric tensors

In this paper, we first introduce the invertibility of even-order tensors and the separable tensors, including separable symmetry tensors and separable anti-symmetry tensors, defined respectively as the sum and the algebraic sum of rank-1 tensors generated by the tensor product of some vectors, say, $v_{1}, v_{2}, \ldots, v_{m}$\mathrm{.}Weshowthatthe$m!$sumrands,eachinform$v_{\sigma(1)}\times v_{\sigma(2)}\times\ldots\times v_{\sigma(m)}$, are linearly independent if $v_1,v_2,\dots ,v_m$ are linearly independent, where $\sigma$ is any permutation on $\{1,2,\dots ,m\}$. We offer a class of tensors to achieve the upper bound for $\text{rank}(A)\le 6$ for all $A\in R^{3\times 3\times 3}$\mathrm{.}Wealsoshowthateach$3\times 3\times 3$ anti-symmetric tensor is separable.