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

For representation spaces $V,W$, the tensor product of group representations acts on product basis vectors throughIts character is the trace. Summing diagonal elements explicitly,No step assumes one-dimensional matrices, so the result applies to both degenerate and nondegenerate representations.

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 $V\otimes V^{\ast }$.