Irreducible representations of $\SO(2)$ and $\SO(3)$

For \(\operatorname{SO}(2)\cong S^1\), every irreducible complex representation is one-dimensional because the group is abelian. They are \[ \chi_n(R_\theta)=e^{in\theta},\qquad n\in\mathbb Z. \] Over \(\mathbb R\), the irreducibles are the trivial one-dimensional representation and, for each \(m\ge1\), the two-dimensional rotation representation \[ R_\theta\longmapsto \begin{pmatrix} \cos(m\theta)&-\sin(m\theta)\\ \sin(m\theta)&\cos(m\theta) \end{pmatrix}. \]

For \(\operatorname{SO}(3)\), use the double cover \[ \operatorname{SU}(2)\to\operatorname{SO}(3). \] The irreducible complex representations of \(\operatorname{SU}(2)\) are \[ \operatorname{Sym}^m(\mathbb C^2),\qquad m\ge0, \] of dimension \(m+1\). The central element \(-I\) acts by \((-1)^m\), so the representation descends to \(\operatorname{SO}(3)\) exactly when \(m=2\ell\) is even. Hence the irreducible complex representations of \(\operatorname{SO}(3)\) are indexed by \(\ell\ge0\) and have dimensions \[ 2\ell+1. \] They may be realized as the degree-\(\ell\) spherical harmonics. These models are defined over \(\mathbb R\), giving the irreducible real representations of the same dimensions.