signature of a permutation
Let be a finite set, and let be the group of permutations of (see permutation group). There exists a unique homomorphism from to the multiplicative group such that for any transposition (loc. sit.) . The value , for any , is called the signature or sign of the permutation . If , is said to be of even parity; if , is said to be of odd parity.
where is the number of pairs such that and . (Such a pair is sometimes called an inversion of the permutation .)
Proof: This is clear if is the identity map . If is any other permutation, then for some consecutive we have and . Let be the transposition of and . We have
and the proposition follows by induction on .
|Title||signature of a permutation|
|Date of creation||2013-03-22 13:29:19|
|Last modified on||2013-03-22 13:29:19|
|Last modified by||rspuzio (6075)|
|Synonym||sign of a permutation|