Signature (permutation)

sign of a permutation

The sign or signature of a permutation of a finite set, which we can identify with $\{1,2,\ldots,n\}$ for some $n$, is a multiplicative map $\epsilon$ from the group of permutations $S_n$ to $\pm 1$. Permutations with sign $+1$ are even and those with sign $-1$ are odd. The sign may be defined in a number of ways. A simple formula is $$ \epsilon(\pi) = \frac{\prod_{1 \le i < j \le n} (x^{\pi(i)}-x^{\pi(j)})}{\prod_{1 \le i < j \le n} (x^i-x^j)} \ .\label{1} $$

The sign of $\pi$ may also be defined by the parity of the number of transpositions which compose $\pi$: this is well-defined since an odd number of transpositions cannot give the identity. A related definition is that the sign is the parity of $n - c$ where $c = c(\pi)$ is the number of cycles (orbits) of $\pi$.

Since $\epsilon$ is a homomorphism from $S_n$ to $C_2 = \{\pm1\}$, the alternating group, or group of even permutations, $A_n$ is the kernel of $\epsilon$ and so a normal subgroup of $S_n$.

The definition of $\epsilon$ may be extended to maps which are not permutations by defining it to be zero. This is consistent with (1) and also with the use of the Kronecker symbol in tensor notation.

References

• Walter Ledermann "Introduction to the theory of finite groups" Oliver and Boyd (1957)