I am trying to get a better grasp of representation theory. I was asking myself "what is the essential difference between representations of some group $G$ and a $KG$ module? How are they related, and what is the distinction?"
What's confusing me is: I can understand matrix representations of a group in a simple way, since they are isomorphic to some permutation group, but what about a module? How do I get things cleared out? I need some insight.
Best Answer
There's essentially no real difference between modules and representations. Think of them as two sides of the same coin.
Given a $\mathbb{K}G$-module $V$, you have a linear action of $G$ on a $\mathbb{K}$-vector space $V$. This in turn gives you a homomorphism from $G$ to $\mathrm{GL}(V)$ (invertible $\mathbb{K}$-linear endomorphisms). Such a homomorphism is a representation. And then this can be turned around. Given a representation, you get an associated module.
Specifically, let $V$ be a $\mathbb{K}G$-module and let $g,h \in G$, $v,w\in V$, and $c\in\mathbb{K}$. Give a name to the map: $v \mapsto g\cdot v$ say: $\varphi(g):V \to V$ (so $\varphi(g)(v)=g \cdot v$). Then $\varphi(g)(v+cw)$ $=g\cdot(v+cw)$ $=g\cdot v+cg\cdot w$ $=\varphi(g)(v)+c\varphi(g)(w)$. Thus $\varphi(g)$ is $\mathbb{K}$-linear. Then because $\varphi(1)$ is the identity map ($1 \cdot v=v$) and $\varphi(g^{-1})(\varphi(g)(v))=g^{-1}\cdot g\cdot v=(g^{-1}g\cdot v=1\cdot v=v$ etc. we get $\varphi(g)$ is an invertible linear map. Therefore: $\varphi:G \to \mathrm{GL}(V)$. Moreover, $\varphi(gh)=\varphi(g)\varphi(h)$ (easy to check) so $\varphi$ is a homomorphism (which we call a representation). Without going into the details, this all reverses.
So $\mathbb{K}G$-modules = representations of $G$ on $\mathbb{K}$-vector spaces.
If you've studied group actions, you've already seen this type of correspondence. Let $G$ act on $X$. Then the map $x \mapsto g \cdot x$ turns out to be a bijection on $X$. Thus if we define $\varphi(g)(x)=g\cdot x$ for all $x\in X$, then $\varphi(g) \in S(X)$ (permutations on $X$). Moreover, $\varphi(gh)=\varphi(g)\varphi(h)$ so $\varphi : G\to S(X)$ is a group homomorphism. We call such things permutation representations. And again this can be reversed. Given a permutation representation: $\varphi:G \to S(X)$, one can define a group action $g \cdot x \equiv \varphi(g)(x)$.
So $G$-action on $X$ = permutation representation of $G$ on $X$.
If you look into other branches of algebra, you'll see this kind of thing over and over again: Lie algebra modules = Lie algebra representations etc.
It's just different points of view. You can either think of "Algebra Thing" acting on "Thing" or a homomorphism from "Algebra Thing" to Maps from "Things to Things".