We show that an isotropic random field on
S
U
(
2
)
SU(2)
is not necessarily isotropic as a random field on
S
3
S^3
, although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on
S
3
S^3
is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree
d
d
is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range
{
−
d
2
,
…
,
d
2
}
\bigl \{-\frac {d}{2},\dots ,\frac {d}{2}\bigr \}
, each of which is isotropic in the sense of
S
U
(
2
)
SU(2)
. Moreover, for a random field of fixed degree, each spin weight appears with the same magnitude, in a sense to be specified.
In addition we will give an overview of the theory of spin weighted functions and Wigner
D
D
-matrices, with the purpose of gathering together many different points of view and adding ours. As a byproduct of this survey we will prove some new properties of the Wigner matrices and a formula relating the operators
ð
ð
¯
\eth \overline {\eth }
and the horizontal Laplacian of the Hopf fibration
S
3
→
S
2
S^3\to S^2
, in the sense of [Bérard Bergery and Bourguignon, Illinois J. Math. 26 (1982), no. 2, 181–200.]