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Is there a symbol $\varepsilon_{ijkl}$ which is zero if any of the four indices are equal and nonzero if the indices are pairwise disjoint?

The indices $i,j,k,l$ should be of the interval $[1,N]$ where $N$ is a large positive integer (in general larger than 4).

Moreover, is it possible to decompose such an object into a sum of objects which only have two indices $\delta_{ij}$ as it is possible for the Levi-Civita symbol?

My goal is to calculate the sum $$ \underset{\text{disjoint}}{\sum_{ijkl}} f_1(i) f_2(j) f_3(k) f_4(l) $$ which I then could write as $$ \sum_{ijkl} \varepsilon_{ijkl} f_1(i) f_2(j) f_3(k) f_4(l) $$ and then hopefully decompose it further with some two-index objetcs $\delta_{ij}$...

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    Six delta functions (one for each combination) might do the job. Or you could simply *define* a new quantity. I can expand to an answer if you'd like. – 2017-02-18
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    @TheCount Are you thinking of something like $\prod^N_{i} \prod_{j – 2017-02-18
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    @KitterCatter yes, exactly. – 2017-02-18
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    Yes an answer would be nice. Thank you. – 2017-02-18

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