1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Simple permutations proof

  1. Mar 16, 2007 #1
    [tex]S_N = \left\{ {\left. {\sigma :\left\{ {1, \ldots ,N} \right\} \to \left\{ {1, \ldots ,N} \right\}} \right|\sigma {\text{ is a bijection}}} \right\}[/tex]
    i.e., the set of all permutations on 'N' values.

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

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

    How do I prove that, for [tex]\sigma ,\pi \in S_N [/tex],
    [tex]{\mathop{\rm sgn}} \left( {\sigma \circ \pi } \right) = {\mathop{\rm sgn}} \left( \sigma \right){\mathop{\rm sgn}} \left( \pi \right) \; ?[/tex]
    Last edited: Mar 16, 2007
  2. jcsd
  3. Mar 17, 2007 #2
    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Simple permutations proof
  1. Simple Ab Alge proof (Replies: 6)