What theorems/examples will make me really understand representation theory?

Question

Okay, so I've been through some basic results on representation theory. I've gone over the proof of Burnside's $pq$ theorem using characters. I've also read though the basics of Lie groups and algebras. However, I still haven't come across a theorem or any examples which set off an "Aha!" moment in which I understand what representations really are and when their use would be appropriate.

For example, when studying group actions, in my opinion the orbit-stabilizer theorem gives me a good idea of what's going on when we study actions on finite groups - I haven't found any such analogue in representation theory.

I suppose part of the problem is that representations are useful in so many distinct ways (finite groups, Lie groups, harmonic analysis, combinatorics, etc.) that I have a hard time synthesizing a coherent picture. Anyone have any good recommendations for books/topic/theorem/examples?

Answer 1

Here is an example from finite groups where a representation shows up "naturally".

Let $G$ be a finite solvable group and let $K/L$ be a chief factor of $G$ (this means that $K$ is a normal subgroup of $G$ and $L$ is a subgroup of $K$ which is maximal among proper subgroups of $K$ which are normal in $G$). Since this corresponds to $K$ being a minimal normal subgroup of the solvable group $G/L$ we see that $K/L$ is elementary abelian, so it is isomorphic to $(\mathbb{F}_p)^n$ for some prime $p$ and some natural number $n$, so it is a vector space over the finite field $\mathbb{F}_p$.

Now $G$ acts on $K/L$ via conjugation, and due to the maximality of $L$, this gives us an irreducible representation of $G$ on the vector space $(\mathbb{F}_p)^n$.

One question is then how this relates to those representations one is usually introduced to, which are over the complex numbers. But it turns out that since $G$ is solvable, we can from the existence of the above representation deduce that there is an irreducible complex character of $G$ of degree at most $n$ and such that a $p$-regular element of $G$ (ie, an element of $G$ whose order is not divisible by $p$) is in the kernel of this irreducible character iff it acts trivially on $K/L$. Further, if the representation from above is faithful (which means that the only $g\in G$ that acts trivially on $K/L$ is the neutral element), then there exists a faithful irreducible complex character of $G$ of degree at most $n$.