Ok, I admit it. I'm confused. I'm a physics student attempting to learn some group theory and topology in my spare time. I was reading about group representations. For example I get that the set of spherical harmonics $Y_{lm}(\theta,\phi)$ form a set of irreducible representations of $SO(3)$. What I don't get is their dimension. For example here (page 144 as it reads on the paper heading) it is stated:
The $Y_{lm}(\theta,\phi)$ form a $(2l+1)$ -dimensional representation of $SO(3)$.
Now, in utilizing the spherical harmonics in physics, I know that I'm working in a three dimensional space. I further know that I can represent any “well behaved” function $f(\theta ,\phi)$ on the unit sphere in $R^3$ in terms of a series of these spherical harmonics (properly weighted with coefficients) like so:
$$f(\theta,\phi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}a_{lm}Y_{lm}(\theta,\phi)$$
It's not lost on me that the dimensionality for a given representation is the same as the number of $m$ values (ie the second summation). I know the function "lives" in a $2$-dimensional space (the unit sphere). So what is going on? Is there a mapping or reference to another space I'm missing? is this a physicist's notational/dictionarial clash with the mathematician's?? Thank you in advance.