this entry is about the signature of a permutation. For other notions of signature see there.
For the symmetric group on elements, the signature is the unique group homomorphism
that sends each transposition , which interchanges the th element with its neighbour and leaves the other elements fixed, to the nontrivial element .
Permutations in the kernel of are called even permutations, and the rest are called odd permutations.
The signature is well-defined.
One way of seeing this is invoking a standard group presentation of where generators for to (representing ) are subject to relations
and checking that the sign applied to both sides of a relation equation gives the same result.
Another is by invoking a tautological representation of on a polynomial algebra ,
and recognizing that for the special polynomial
we have, for each permutation , either or . (The polynomial , which is invariant under the action, is called the discriminant.)
There are various means for computing the signature (also called sign) of a permutation.
The definition itself suggests one method: if we linearly order the set by , then we can exhibit a permutation by a string diagram and simply count the number of crossings ; then we have
Each crossing corresponds to a pair of elements such that , called an inversion.
Another method which does not depend on choosing a total order is to exhibit a permutation through its cycle decomposition. Each cycle of period contributes a sign , and the overall sign is the product of these contributions taken over all the cycles.