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 c(\lambda) of a Young diagram \lambda is the sum \sum_{j=1}^k\sum_{i=1}^{\lambda_{j}}(i-j), where \lambda=(\lambda_{1},...,\lambda_{k}) is a partition of \lambda. Let C=\sum_{i<j}(ij)\in\mathbb{C}[S_{n}] be the sum of all transpositions. Show that C acts on the Specht module V_{\lambda} by multiplication by c(\lambda).

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.