On Wikipedia, elements of Wigner's D-matrix are defined as
$$D_{m'm}^{j}(\alpha,\beta,\gamma)=\langle jm'|e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}|jm\rangle=e^{-im'\alpha}d_{m'm}^j (\beta)e^{-im\gamma }$$ where $d_{m'm}^j (\beta)=\langle jm'|e^{-i\beta J_y}|jm\rangle$ is the Wigner (small) d-matrix element. According to the documentation, such Wigner D-functions are implemented in Mathematica via the WignerD
command: "WignerD[{j, m1, m2}, ψ, θ, ϕ]
gives the Wigner D-function $D_{m_1m_2}^j(\psi,\theta,\phi)$."
However, something seems off. For instance, according to Wikipedia one has
$$D_{1/2,-1/2}(0,\beta,0)=d_{1/2,-1/2}(\beta)=-\sin(\beta/2).$$
whereas Mathematica's WignerD
command yields
In[1]:= WignerD[{1/2, 1/2, -1/2}, 0, β, 0]
Out[1]= Sin[β/2].
In[2]:= WignerD[{1/2, -1/2, 1/2}, 0, β, 0]
Out[2]= -Sin[β/2].
So Mathematica seems to disagree with Wikipedia as to the overall sign (or, equivalently, the order of $m',m$). This presumably reflects some convention for how the Wigner D-functions are defined in Mathematica, but I haven't seen this documented anywhere. Can anyone clear up the confusion?