So my professor has given me the problem of examining the representations of $SO(3)$ in $V = \mathbb{C}[x,y,z]$. The thing is that he's now gone away for a conference trip, and I have absolutely no idea where to start.
I don't have a super solid background in representation theory; I've read the first part of Serre's Linear Representations of Finite Groups, and I know the basic concepts of compact groups and the Haar measure. To me this doesn't seem like enough in order to tackle this problem, but he insists that it is.
Is it possible for someone to help me get started? As noted earlier, I'm totally lost...
Thanks in advance!