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We show that an isotropic random field on $SU(2)$ is not necessarilyisotropic as a random field on $S^3$, although the two spaces can beidentified. The ambiguity is due to the fact that the notion of isotropy on agroup and on a sphere are different, the latter being much stronger. We showthat any isotropic random field on $S^3$ is necessarily a superposition ofuncorrelated random harmonic homogeneous polynomials, such that the one ofdegree $d$ is necessarily a superposition of uncorrelated random spin weightedfunctions of every possible spin weight in the range$\{-\frac{d}{2},\dots,\frac{d}{2}\}$, each of which is isotropic in the senseof $SU(2)$. Moreover, for a random field of fixed degree, each spin weightappears with the same magnitude, in a sense to be specified.  In addition we will give an overview of the theory of spin weighted functionsand Wigner $D$-matrices, with the purpose of gathering together many differentpoints of view and adding ours. As a byproduct of this survey we will provesome new properties of the Wigner matrices and a formula relating the operators$\eth\overline{\eth}$ and the horizontal Laplacian of the Hopf fibration$S^3\to S^2$.

Isotropic random spin weighted functions on $S^2$ vs isotropic random fields on $S^3$