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Homesignature of a permutation

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# 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
$x<y$ 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 $a<b$ and $g(a)>g(b)$. Let $h\in G$
be the transposition of $a$ and $b$. We have

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

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

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

## Mathematics Subject Classification

03-00*no label found*05A05

*no label found*20B99

*no label found*

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