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In N-dimensions Levi-Civita symbol is defined as:

\begin{align}

\varepsilon_{ijkl\dots}=

\begin{cases}

+1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\

-1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\

0 & \mbox{otherwise}

\end{cases}

\end{align}

I found the following expression for the product of two Levi-Civita symbols when there are no dummy indices (i_1,...,i_n,j_1,...,j_n are in {1,...,n}):

\begin{align}& \varepsilon_{i_1 \dots i_n} \varepsilon^{j_1 \dots j_n} = n! \delta^{j_1}_{[ i_1} \dots \delta^{j_n}_{i_n ]} &&\\& \end{align}

But I could not find its proof through literature and also I was failed to prove it!

Can you please help me? Thanks!

Best Regards,

Ali

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# Product of Two Levi-Civita Symbols in N-dimensions

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