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Help solving a problem on representation theory

  1. May 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Hi all,
    I'm having to solve a few exercises from the book "Introduction to representation theory" (Etingof, Goldberg,...), and I am stuck on an exercise. In the book it's number 5.16.2:
    The content [itex]c(\lambda)[/itex] of a Young diagram [itex]\lambda[/itex] is the sum [itex]\sum_{j=1}^k\sum_{i=1}^{\lambda_{j}}(i-j)[/itex], where [itex]\lambda=(\lambda_{1},...,\lambda_{k})[/itex] a partition of [itex]\lambda[/itex]. Let [itex]C=\sum_{i<j}(ij)\in\mathbb{C}[S_{n}][/itex] be the sum of all transpositions. Show that [itex]C[/itex] acts on the Specht module [itex]V_{\lambda}[/itex] by multiplication by [itex]c(\lambda)[/itex].

    I've been able to work this out with a few examples, but I don't really know how to get a proof.

    Any help is much appreciated,
    Thank you.
     
  2. jcsd
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