My question is simply wrote on the title. (I'm using Einstein's contraction rule.)
In the case of three dimensions, I can construct the Levi-Civita-like tensor as follows. \begin{align} e_{ijk} = \begin{cases} 1 \quad for \ i=j=k \\ 1 \quad for \ (i,j,k) \in even \ or \ odd \ permutations \ of \ (1,2,3) \\ 0 \quad otherwise \end{cases}. \end{align} This satisfies the equation \begin{align} e_{ijk} e_{lmk} = \delta_{il} \delta_{jm}. \end{align} Is it possible to construct such tensors in higher dimensions? And if it is, how can I get some of these practically?
Thank you in advance for any suggestions.
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$\begingroup$That equation is incorrect; I'm not used to this notation but I think it's actually
$$e_{ijk} e_{lmk} = \delta_{il} \delta_{jm} + \delta_{im} \delta_{jl}.$$
In fact a tensor with this property does not exist in any dimension at least $2$. The reason is that it would imply the existence of a pair of linear maps $f : V \otimes V \to V$ and $g : V \to V \otimes V$ (where $V$ is a vector space of the given dimension) such that the composite
$$g \circ f : V \otimes V \to V \otimes V$$
is the identity. But this is impossible when $\dim V \ge 2$ because $\dim V \otimes V = (\dim V)^2 > \dim V$, and so neither $f$ nor $g$ can have full rank.
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