I'm having a little trouble understanding (intuitively or otherwise) what is meant by a module over a group algebra FG (F a field, G a finite group) and how it relates to transformations.
For example, how do I go about thinking about this question:
Identify with the subgroup of which fixes 4 ( is of course the symmetric group on letters). Then acts on by conjugation (???). Find the character of the linearization as a -module, and determine the multiplicities of each simple -module in .
I would be very very grateful for any light anyone could shed on this question; especially any general comments about FG-modules and representations.
Thanks in advance!