Permutational–symmetry-adapted powers of representations of the point groups deduced from the unitary group

The construction of symmetrized powers of representations of the point groups can be deduced by decomposing the appropriate representation of the unitary group into the representations of the group of the sphere, and then mapping these representations onto the point group. The method is generally simpler than the traditional method based on the character tables of the symmetric group. As an example, the symmetry-adapted powers (N) of the representations of the icosahedral group are presented for 2≤N≤5.