Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:
1) The character table of a group is packs a lot of information about the group and is concise.
2) It is practically/computationally nice to have explicit matrices that model a group.
But there are for sure deeper things that I am missing. I can understand why one would want to study group actions (the axioms for a group beg you to think of elements as operators), but why look at group actions on vector spaces? Is it because linear algebra is so easy/well-known (when compared to just modules, say)?
I am also told that representation theory is important in quantum mechanics. For example, physics should be $SO(3)$ invariant and when we represent this on a Hilbert space of wave-functions, we are led to information about angular momentum. But this seems to only trivially invoke representation theory since we already start with a subgroup of $GL(n)$ and then extend it to act on wave functions by $\psi(x,t) \mapsto \psi(Ax,t)$ for $A$ in $SO(n)$.
This http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory wikipedia article claims that if our physical system has $G$ as a symmetry group, then there is a correspondence between particles and representations of $G$. I'm not sure if I understand this correspondence since it seems to be saying that if we act an element of G on a state that corresponds to some particle, then this new state also corresponds to the same particle. So a particle is an orbit of the $G$ action? Anyone know of good sources that talk about this?