Definition:Sign of Permutation
Definition
Let $n \in \N$ be a natural number.
Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.
Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers.
Let $\pi$ be a permutation of $\N_n$.
Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$.
Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as:
- $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$
The sign of $\pi \in S_n$ is defined as:
- $\map \sgn \pi = \begin {cases}
\dfrac {\Delta_n} {\pi \cdot \Delta_n} & : \Delta_n \ne 0 \\ 0 & : \Delta_n = 0 \end {cases}$
Also denoted as
Some sources use $\map \epsilon \pi$ for $\map \sgn \pi$.
In physics and applied mathematics, the symbol $e_{i j k}$ can often be found for this concept, referred to as the alternating symbol, defined as:
- $e_{i j k} = \begin{cases}
1 & : \text {if $\tuple {i, j, k}$ is an even permutation of $\tuple {1, 2, 3}$} \\ -1 & : \text {if $\tuple {i, j, k}$ is an odd permutation of $\tuple {1, 2, 3}$} \\ 0 & : \text {if any two of $\set {i, j, k}$ are equal} \end{cases}$
Also known as
The sign of a permutation is also known as its signature or signum.
However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ signum is not recommended, in order to keep this concept separate from the signum function on a set of numbers.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image: Example $142$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): signature: 1.
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Definition $9.15$
- 1980: A.J.M. Spencer: Continuum Mechanics ... (previous) ... (next): $2.1$: Matrices: $(2.10)$