Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 49 / b / ii / Solution

For a real-character irreducible molecular representation with character $\chi$, the multiplicity of the fully symmetric representation in its tensor square isby character orthogonality. The identity uses the real-character assumption appropriate to the usual real irreducible representations intended here. For a general complex irreducible representation, the trivial representation in a tensor square has multiplicity one precisely when it is self-dual, and zero otherwise, by Schur's lemma. The universally valid tensor product is $V\otimes V^{\ast }$.