For each value of j, the 2 j +1 kets form a basis for an irreducible representation (irrep) of SO(3)/SU(2) over the complex numbers. Given two irreps, the tensor direct product can be reduced to a sum of irreps, giving rise to the Clebcsh-Gordon coefficients, or by reduction of the triple product of three irreps to the trivial irrep 1 giving rise to the 3j symbols.