\begin{definition}[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
:$0$ 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).
\end{definition}