# signature of a permutation

Let $X$ be a finite set^{}, and let $G$ be the group of permutations^{} of $X$ (see
permutation group^{}). There exists a unique homomorphism^{} $\chi $ from $G$ to the
multiplicative group^{} $\{-1,1\}$ such that $\chi (t)=-1$ for any transposition^{}
(loc. sit.) $t\in G$. The value $\chi (g)$, for any $g\in G$, is called the
*signature ^{}* or

*sign*of the permutation $g$. If $\chi (g)=1$, $g$ is said to be of even

*parity*; if $\chi (g)=-1$, $g$ is said to be of odd parity.

Proposition^{}: If $X$ is totally ordered by
a relation^{} $$, then for all $g\in G$,

$$\chi (g)={(-1)}^{k(g)}$$ | (1) |

where $k(g)$ is the number of pairs $(x,y)\in X\times X$ such that
$$ and $g(x)>g(y)$. (Such a pair is sometimes called an *inversion ^{}*
of the permutation $g$.)

Proof: This is clear if $g$ is the identity map^{} $X\to X$.
If $g$ is any other permutation, then for some
*consecutive* $a,b\in X$ we have $$ and $g(a)>g(b)$. Let $h\in G$
be the transposition of $a$ and $b$. We have

$k(g\circ h)$ | $=$ | $k(g)-1$ | ||

$\chi (g\circ h)$ | $=$ | $-\chi (g)$ |

and the proposition follows by induction^{} on $k(g)$.

Title | signature of a permutation |

Canonical name | SignatureOfAPermutation |

Date of creation | 2013-03-22 13:29:19 |

Last modified on | 2013-03-22 13:29:19 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 03-00 |

Classification | msc 05A05 |

Classification | msc 20B99 |

Synonym | sign of a permutation |

Related topic | Transposition |

Defines | inversion |

Defines | signature |

Defines | parity |

Defines | even permutation |

Defines | odd permutation |