Chapter 4 - Representations of Groups — Principal Ideas

Abstract

This chapter describes the concept of the representation of a group. The representation is described as being faithful if the mapping is one-to-one. For a Lie group it is necessary to supplement the definition by the requirement that the homomorphic mapping must be continuous. For a connected linear Lie group, the matrix elements of the representation must be the continuous functions of the parameters. Every group has an infinite number of different representations, but these can be formed out of certain irreducible representations. It is found that vector spaces and inner product spaces play an important part in representation theory. All the point groups and space groups of solid state physics are finite. The rotation groups in three dimensions and the internal symmetry groups of elementary particles are compact Lie groups. A noncompact Lie group that is not simple may possess both unitary representations and representations that are not equivalent to unitary representations. The reducible and irreducible representations are also elaborated.