The Clebsch-Gordan coefficients relate the direct product of two irreps to
a sum of irreps:
direct product(D(u),D(v)=Sum(direct sum,AsD(s))
where the coefficients As are given by:
As=
the Clebsch-Gordan coefficients can be regarded as forming a unitary matrix
M
Example:
For SU(2):
direct product(D(j1),D(j2))=Sum(j=|j1-j2|,j1+j2,D(j))