Definition:Permutation Symbol

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


 * $$\varepsilon_{ijk\dots} \ $$,


 * $$\varepsilon^{ij\dots}_{kl\dots} \ $$,


 * $$\varepsilon^{ijk\dots} \ $$.

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