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 of a Young diagram is the sum , where is a partition of . Let be the sum of all transpositions. Show that acts on the Specht module by multiplication by .
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,