How Does the Sign of a Permutation Product Relate to Its Components?

[bomba923]

Consider
S_N = \left\{ {\left. {\sigma :\left\{ {1, \ldots ,N} \right\} \to \left\{ {1, \ldots ,N} \right\}} \right|\sigma {\text{ is a bijection}}} \right\}
i.e., the set of all permutations on 'N' values.

Define
\Delta \left( {x_1 , \ldots ,x_N } \right) = \prod\limits_{i < j} {\left( {x_i - x_j } \right)}
and, for \sigma \in S_N,
\sigma \left( \Delta \right)\left( {x_1 , \ldots ,x_N } \right) = \prod\limits_{i < j} {\left( {x_{\sigma \left( i \right)} - x_{\sigma \left( j \right)} } \right)}

Also, define {\mathop{\rm sgn}} : S_N \to \left\{ {\pm 1} \right\} as
{\mathop{\rm sgn}} \left( \sigma \right) = \left\{ \begin{array}{l}<br /> 1,\;\sigma \left( \Delta \right) = \Delta \\ <br /> - 1,\;\sigma \left( \Delta \right) = - \Delta \\ <br /> \end{array} \right.

How do I prove that, for \sigma ,\pi \in S_N,
{\mathop{\rm sgn}} \left( {\sigma \circ \pi } \right) = {\mathop{\rm sgn}} \left( \sigma \right){\mathop{\rm sgn}} \left( \pi \right) \; ?

 

[MathematicalPhysicist]

sgn(sigma o pi)=1
then sigma(pi(delta))=delta
if pi(delta)=delta then sigma(delta)=delta so both sgn are 1.
if pi(delta)=-delta sigma(-delta)=delta, which is the same as sigma(delta)=-delta, you can see it by obserivng that sigma(delta)+sigma(-delta)=0.
the same goes when sgn (sigma(pi))=-1.