Signature

The signature (or sign) of a permutation $g\in S_n$ is

$$\sigma (g)=\prod _{i<j}\frac{g(i)-g(j)}{i-j}\in \{\pm 1\},$$

or equivalently, expressed via cycle decomposition: if $g=c_1{c}_2\cdots c_m$ in disjoint cycles with $|c_k|=r_k$,

$$\boxed{\sigma (g)=\prod _{k=1}^m(-1{)}^{r_k-1}.}$$

So a cycle of length $r$ has signature $(-1{)}^{r-1}$ - odd-length cycles are even, even-length cycles are odd.

Easier Mental Recipe

Signature $=+1$ iff the number of even-length cycles in the disjoint-cycle decomposition is even. (Each even-length cycle contributes a factor of $-1$.)

Properties

• $\sigma$ is a Homomorphism $S_n\to \{\pm 1\}$ (multiplicative group, $\cong C_2$).
• $\sigma$ is surjective for $n\ge 2$ (any transposition has signature $-1$).
• $\ker (\sigma )=A_n$, the Alternating group.
• $\sigma (g^{-1})=\sigma (g)$ (cycles reverse but lengths preserved).
• $\sigma (gh{g}^{-1})=\sigma (h)$ - conjugation preserves signature (preserves cycle structure).

Even and Odd Permutations

• Even ($\sigma =+1$): in $A_n$.
• Odd ($\sigma =-1$): not in $A_n$ - the “other half” of $S_n$.

In Action

A transposition $(ij)$ is a 2-cycle, signature $-1$. Any permutation can be written as a product of transpositions; the parity of the number of transpositions equals the signature.

Examples

• $\sigma (e)=+1$.
• $\sigma ((12))=-1$.
• $\sigma ((123))=+1$ (3-cycle has length 3, $(-1{)}^2=+1$).
• $\sigma ((123)(45))=(+1)(-1)=-1$.
• $\sigma ((12)(34)(56)(78))=(-1{)}^4=+1$.

Determinant Connection

The signature equals the determinant of the corresponding permutation matrix: $\sigma (g)=\det (P_g)$. This is the geometric origin of the sign - it tracks orientation reversal under permutation of basis vectors.