Help solving a problem on representation theory

[pap218]

Homework Statement

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 $\lambda=(\lambda_{1},...,\lambda_{k})$ 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.

 

[lippyka]

Homework Equations C=\sum_{i<j}(ij)\in\mathbb{C}[S_{n}]The Attempt at a Solution I'm not sure how to solve this problem. I tried looking online for some help but couldn't find anything that was helpful.