# Definition:Permutation Symbol

Jump to navigation
Jump to search

It has been suggested that this page or section be merged into Definition:Sign of Permutation.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## 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
- $0$ 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).