My supervisor has asked me if 'Wigner matrices' are «rotation matrices written in spherical coordinates». I think he's asking if a point (read: a 3-tuple of spherical coordinates) on a unit sphere can be rotated by applying the appropriate Wigner matrix.
I have heard from Wikipedia that «Wigner matrices are the ordinary rotation matrices written in the simultaneous eigenbases of $J^2$ and $J_z$». That seems to mean that if $d$ is the dimension of the lie algebra so($d$), $j$ is the quantum number of total angular momentum, $m$ is the number for angular momentum about the $z$-direction and $v_{m,j} \in \mathbb{R}^d$ denotes a basis such that
$$J_z v_{m,j} = m\,v_{m,j} \qquad J^2 v_{m,j} = j(j+1) \, v_{m,j},$$
then the Winger matrix is $D^j_{m',m} = Qv_{m',j} \cdot v_{m,j}$. Here, $\cdot$ refers to the ordinary dot product and $Q \in O(3)$ is a rotation matrix.
So it seems I can find the answer to my problem if I can get the details of that basis $\{v_{m,j}\}$ (as a minimum, the formula for it). But the problem is that it seems very difficult to learn what that basis looks like. I haven't found any formulas in any books or on the internet. Do you know the details of this basis (such as the formula for it) and could share it with the community?
d is the dimension, 2j+1, of the representation of the algebra so(3), not so(d)! Any decent book on QM, such as Sakurai's and Napolitano's, describes this basis, $_{,}$≡|⟩, as your WP link does.
The Casimir operator $J^2 = J_x^2 + J_y^2 + J_z^2 $ commutes with all three elements of the Lie algebra, $[J_x,J_y] = i J_z,\quad [J_z,J_x] = i J_y,\quad [J_y,J_z] = i J_x$, so it may be diagonalized together with $J_z$ thereby defining this spherical basis.
In this basis, $J^2\propto 1\!\!1$, the $(2j+1)$-dimensional identity matrix. $j$ is the label of the representation, while $m$ is the label of each component thereof, ranging from $-j$ to $j$, in integer steps; so there are $2j+1$ such components, acted upon by $(2j+1)\times(2j+1)$ Lie algebra matrices, and hence Lie group elements, the rotation matrices. (j = 0, 1/2, 1, 3/2, 2, ... for su(2), and = 0, 1, 2, ... for so(3). Since the Lie algebras are the same, both groups are dubbed the "rotation group".)
A rotation group element can be written as $\mathcal{R}(\alpha,\beta,\gamma) = e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z},$ where α, β, γ are Euler angles. How is it represented by matrices?
The Wigner D-matrix is a unitary $(2j+1)\times (2j+1)$ square matrix in this spherical basis with elements $$D^j_{m'm}(\alpha,\beta,\gamma)\equiv\langle jm'| \mathcal{R} (\alpha,\beta,\gamma)|jm\rangle= e^{im'\alpha} D^j_{m'm}(0,\beta,0)e^{-im\gamma},\\ D^j_{m'm}(0,\beta,0)=\langle jm'|e^{-i\beta J_y}|jm \rangle \equiv d^j_{m'm} . $$
The second line is the little-$d^j_{m'm}$, for which there is a compact expression in the WP article, as well as a brute listing for the smallest irreducible representations: j= 1/2; 1; 3/2; 2.
You may also whip them out by exponentiating the $J_y$ algebra elements in these irreps, even though ladder operator methods are more efficient.
Most decent books on angular momentum, Edmonds (ISBN-13:978-0691079127), Rose (ISBN-13:978-0486684802), Biedenharn & Louck(ISBN-13:978-0521302289), etc, cover them in excruciating detail...