As indicated at the start of the previous chapter, significant conclusions can often be drawn about a physical system simply from the study of its symmetry properties. That chapter was devoted to setting up a formal mathematical basis, group theory, with which to describe and classify such properties; the current chapter shows how to implement the consequences of the resulting classifications and obtain concrete physical conclusions about the system under study. The connection between the two chapters is akin to that between working with coordinate-free vectors, each denoted by a single symbol, and working with a coordinate system in which the same vectors are expressed in terms of components.

The ‘coordinate systems’ that we will choose will be ones that are expressed in terms of matrices; it will be clear that ordinary numbers would not be sufficient, as they make no provision for any non-commutation amongst the elements of a group. Thus, in this chapter the group elements will be represented by matrices that have the same commutation relations as the members of the group, whatever the group's original nature (symmetry operations, functional forms, matrices, permutations, etc.). For some abstract groups it is difficult to give a written description of the elements and their properties without recourse to such representations. Most of our applications will be concerned with representations of the groups that consist of the symmetry operations on molecules containing two or more identical atoms.