I'm given a field $k$, a finite group $G$ and a set $S$ which $G$ acts on transitively. I'm then told to consider the permutation module $M = kS$.
My first problem is understanding what the permutation module actually is? My guess is that it is the module with basis $\{b_s : s \in S\}$ and then multiplication coming from $G$ where $g.b_s = b_{gs}$ which is then extended linearly. But then I'm not sure how $k$ plays a part in this?
I'm then told to consider the subsets:
- $M_1 = k (\sum b_s)$ where the sum is over all $s \in S$
- $M_2 = \big\{\sum \lambda_s b_s : \sum \lambda_s = 0$
Now the question asks me to determine the vector space dimensions of $M_1$ and $M_2$, however if I consider these as $k$-vector spaces which is my guess? Then the dimension of $M_1$ is just 1 because it has $\sum b_s$ is a basis and then dimension of $M_2$ is $|S|-1$ because all but one of the scalars are free by the condition on their sum.
I'm really not sure if this is the right interpretation of these objects and would appreciate some help in the right direction. I'm particularly skeptical because I am later asked to determine the corresponding representations to $M_1$ and $M/M_2$ but then these things are both 1 dimensional and then I would get that both the representations are trivial? (Just because say for $M_1$ I would have $p: G \to \operatorname{End}_k(M_1)$ would be given by $p(g) (v) = p(g) (\lambda \sum b_s) = \lambda \sum b_gs = \lambda \sum b_g$ because G acts transitively)
Apologies for the long post, but I've struggled to find any good references on this topic and am having difficulty understanding some of these ideas. I appreciate any help!