Reducible and irreducible representations

Abstract

We need to know the relationship between any arbitrary reducible representation and the irreducible representations of that point group. As was mentioned in the last chapter, there is no limit to the order of a representation; that is, it may consist of matrices of any size. Some of the reducible representations that are of use in, for example, vibrational spectroscopy are of very large dimensions. Frequent use is made of the representations generated by placing three Cartesian coordinate vectors on each atom in a molecule. For an N-atomic molecule, this will give a representation consisting of 3N × 3N matrices; for example, for benzene, the matrices will be of dimension 36 × 36. In order to handle and analyse such matrices, it is essential to reduce them to more manageable sizes; that is, to reduce them eventually to those of the irreducible representations, which are the representations of the smallest possible dimensions for any point group. How can this be done?