Signum

algebra

Definition

Voreichen

Signum

The signum of a number $x$ is defined as:

$$\mathrm{sgn}(x)=\{\begin{array}{ll}1& \text{if }x>0\\ 0& \text{if }x=0\\ -1& \text{if }x<0\end{array}$$

Permutation

Let $\pi \in S_n$ be a permutation, then the signum is given by:

$$\mathrm{sgn}(\pi )=\prod _{1\le i<j\le n}\frac{\pi (j)-\pi (i)}{j-i}$$

Even Permutation

A permutation $\pi \in S_n$ is called even if $\mathrm{sgn}(\pi )=1$.

Examples:

• (12)(34)
• (13)(24)(5)
• (13)(24)(5)(6)(7)

The permutations are written in cycle notation.

Odd Permutation

A permutation $\pi \in S_n$ is called odd if $\mathrm{sgn}(\pi )=-1$.

Examples:

• (12)(3)
• (12)(6)(34)(52)
• (12)(6)(34)(52)(7)

The permutations are written in cycle notation.

Kernel

$$\ker (\mathrm{sgn})=\{\pi \in S_n\mid \mathrm{sgn}(\pi )=1\}=:{\mathbf{A}}_n$$

where ${\mathbf{A}}_n$ is called the alternating group.