Definition:Permutation Symbol

Definition
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, and $0$ if the list of elements is not a permutation (i.e. contains a repeated value).

Frequently, the permutation will be explicit, eg, written


 * $\varepsilon_{i j k \ldots}$


 * $\varepsilon^{i j \ldots}_{kl \ldots}$


 * $\varepsilon^{i j k \ldots}$

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

Also see

 * Definition:Sign of Permutation