>[!quote] In a Nutshell >[[Group Action|Irreducible matrix representation]] of the [[Group|group]] $SO(3)$. Higher order representations are needed for the transformation of more complex objects. While vectors simply transform with the regular $3D$ matrices, e.g. [[Tensors|rank-2 tensors]] (that are symmetric and traceless) require the $5D$ transformation. --- >[!info] Definition >The (**real**) Wigner-D [[Matrices|matrices]] of type or frequency $\ell$ are the [[Group Action|irreducible]] matrix representations of $SO(3)$. They consist of $(2 \ell +1)\times (2 \ell + 1)$- dimensional matrices $\mathbf{D}^{(\ell)}(\mathbf{R})=[D_{mn}^{(\ell)}(\mathbf{R})]^{\ell}_{m,n=-\ell},$where the Wigner-D functions $D_{mn}^{\ell}$ form an orthogonal basis for the $l_2$ functions on $SO(3)$. The matrices naturally act on $(2 \ell +1)$-dimensional vectors, which we denote **steerable vectors of type $\ell$**. naturally, the associated [[Vector Space|vector space]] is denoted **steerable vector space of type $\ell$**. - **Type $0$**: A scalar cannot contain directional information, therefore the $1\times1$ matrix is just the identity$\mathbf{D}^{(0)}(\mathbf{R})\mathbf{v}=1\mathbf{v},$they are invariant under all possible rotations. >[!brainwaves] Intuition >Can be seen as the constant part of a [[Fourier Series and Transform|fourier transform]], frequency $0$. This denotes a constant function on the sphere. **Rotating it doesn't change it**. - **Type $1$**: A regular three-dimensional vector that transforms directly via the rotation matrix $\mathbf{R}\in SO(3)$. For a specific choice of [[- A Hitchhiker's Guide to Rotation Representations -|euler angles]] $\alpha, \beta, \gamma$ (there are many forms, also complex ones, because of trig identitites !) they may look like$\begin{align}R^1(\alpha, \beta, \gamma) &= \begin{pmatrix} \cos(\alpha)\cos(\beta)\cos(\gamma) - \sin(\alpha)\sin(\gamma) & -\cos(\alpha)\cos(\beta)\sin(\gamma) - \sin(\alpha)\cos(\gamma) & \cos(\alpha)\sin(\beta) \\ \sin(\alpha)\cos(\beta)\cos(\gamma) + \cos(\alpha)\sin(\gamma) & -\sin(\alpha)\cos(\beta)\sin(\gamma) + \cos(\alpha)\cos(\gamma) & \sin(\alpha)\sin(\beta) \\ -\sin(\beta)\cos(\gamma) & \sin(\beta)\sin(\gamma) & \cos(\beta) \end{pmatrix} \\ &=\begin{pmatrix} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33} \end{pmatrix}\end{align}.$These types are invariant under rotations by $2 \pi$. >[!brainwaves] Intuition >Directional information like an arrow. Rotating it changes is 1:1, only invariant when rotated fully. - **Type $2$** A $5$-dimensional representation is for example given by$R^2(\alpha, \beta, \gamma) = \begin{pmatrix} \cos(\alpha)\cos(\beta)^2\cos(\gamma) - \sin(\alpha)\sin(\gamma) & -\cos(\alpha)\cos(\beta)^2\sin(\gamma) - \sin(\alpha)\cos(\gamma) & \cos(\alpha)\cos(\beta)\sin(\beta) & \cos(\alpha)\sin(\beta)^2\cos(\gamma) & -\cos(\alpha)\sin(\beta)^2\sin(\gamma) \\ \sin(\alpha)\cos(\beta)^2\cos(\gamma) + \cos(\alpha)\sin(\gamma) & -\sin(\alpha)\cos(\beta)^2\sin(\gamma) + \cos(\alpha)\cos(\gamma) & \sin(\alpha)\cos(\beta)\sin(\beta) & \sin(\alpha)\sin(\beta)^2\cos(\gamma) & -\sin(\alpha)\sin(\beta)^2\sin(\gamma) \\ -\cos(\beta)\sin(\beta)\cos(\gamma) & \cos(\beta)\sin(\beta)\sin(\gamma) & \cos(\beta) & \sin(\beta)\cos(\gamma) & -\sin(\beta)\sin(\gamma) \\ -\sin(\beta)^2\cos(\gamma) & \sin(\beta)^2\sin(\gamma) & \cos(\beta)\sin(\beta) & \cos(\beta)^2\cos(\gamma) & -\cos(\beta)^2\sin(\gamma) \\ \sin(\beta)^2\cos(\gamma) & -\sin(\beta)^2\sin(\gamma) & -\cos(\beta)\sin(\beta) & \cos(\beta)^2\cos(\gamma) & \cos(\beta)^2\sin(\gamma) \end{pmatrix}.$These types are invariant under rotations by $\pi$. >[!brainwaves] Intuition >Orientational information like above, but with additional structure / symmetry $v=-v$. i.e. when rotating by $\pi$ (two axes of symmetry) - **Type $\ell$**: higher order elements represent higher frequencies and are invariant under rotations by $\theta =\frac{2 \pi}{\ell}.$ >[!brainwaves] Intuition >More expressive with $\ell$ axes of symmetry. --- >[!success] Theorem >The Wigner -D functions $D_{mn}^{(\ell)}\colon SO(3)\rightarrow\mathbb{R}$ form a complete orthogonal basis for functions on $SO(3)$, meaning every function can be represented in such an $SO(3)$ [[Fourier Series and Transform|Fourier series]] $f(\mathbf{R})=\sum\limits_\ell \sum\limits_{m=-\ell}^\ell \sum\limits_{n=-\ell}^{\ell}\hat{f}_{mn}^{(\ell)} D_{mn}^\ell (\mathbf{R})=\sum\limits_\ell \text{trace}\big(\hat{f}^{(\ell)} \mathbf{D}^{(\ell)}(\mathbf{R}^{-1}) \big),$where the summations are performed by the [[Matrices|matrix]] product and trace operator. - This means that e.g. for $\ell=1$, we have that$\begin{align}R_{11}R_{21}+R_{12}R_{22}+R_{13}R_{23}=0.\end{align}$The same holds for every row and column. The central columns $D_{:0}$ are invariant to rotations around chosen reference axes (compare to examples: they only contain $2$ of the $3$ angles). Mathematically, we say$\exists_{\alpha \in [0, 2\pi)}\colon D_{m0}^{(\ell)}(\mathbf{R}\mathbf{R}_{\alpha})=D_{m0}^{(\ell)}(\mathbf{R}).$It can be shown that the subset of these $SO(2)$-invariant basis functions coincide with the [[Spherical Harmonics|spherical harmonics]] $Y \colon \mathbb{S}^{2} \rightarrow \mathbb{R}$ on the [[Quotient Group|quotient group]] / [[Group Action|homogeneous space]] $\mathbb{S}^{2}=SO(3)/SO(2)$: $Y^{(\ell)}(\mathbf{n}_{\beta, \gamma})\sim D_{m0}^{(\ell)}(\mathbf{R}_{\alpha,\beta,\gamma}).$This means that if I want to rotate a function on a sphere, I can transform it via a Fourier transform, shift it with the Wigner-D matrices and transform it back$f(\mathbf{Rn})=[\mathcal{F}_{\mathbb{S}^{2}}^{-1}\mathbf{D}(\mathbf{R})\mathcal{F}_{\mathbb{S}^{2}}f](\mathbf{n})$