Definition:Permutation Symbol

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The permutation symbol $\varepsilon$ of a permutation $P$ of a set of elements is defined as:

$+1$ for even permutations (permutations that are an even number of pair swaps)
$-1$ for odd permutations
$0$ if the list of elements is not a permutation (that is, contains a repeated value).

Frequently, the permutation will be explicit, for example:

$\varepsilon_{i j k \ldots}$
$\varepsilon^{i j \ldots}_{k l \ldots}$
$\varepsilon^{i j k \ldots}$

This notation is especially useful when raising and lowering indices (that is, converting between forms and vectors).

Also see