The basic ideas of group representation theory were presented in Chapter 1. For finite groups and for compact simple Lie groups, every representation can be assumed to be unitary. The basic building blocks, or units, of representation theory are then the unitary irreducible representations (UIR’s), which are all finite dimensional. The UIR's of the permutation groups Sn , and of compact simple Lie groups, have been described in Chapters 2 and 5, respectively.

The two regular representations of any group G of interest have special properties which we will now draw upon: (i) The representation space is the space of all complex valued functions on G , made into a Hilbert space with a suitably defined inner product; (ii) there are two mutually commuting regular representations, the left and the right; (iii) in each of them, upon complete reduction, each UIR appears with multiplicity equal to its dimension. Thus for example forG = SU (2), in each regular representation the spin j UIR D (j), j = 0, 1/2, 1, … , appears (2j + 1) times.

In the particular case when G is abelian, for instance G = SO (2), each UIR is one dimensional, the two regular representations coincide, and each of them is a multiplicity free direct sum of all the UIR’s.

Now we turn to general UR's of G . When a UR is fully reduced and expressed as a direct sum of UIR’s, each distinct UIR occurs with some multiplicity. Thus the UR as a whole is in principle completely determined up to unitary equivalence by these multiplicities. However, certain UR's may have special significance, reflecting the way they are constructed, and so merit special attention. We will study the following interesting UR's – one which we call the Schwinger representation of a group, and others obtained by an elegant ‘process of induction’ from UIR's of various Lie subgroups H in G .We conclude the chapter with a brief description of generalised coherent states based on UIR's of Lie groups. These are important in many physical contexts, and their analysis makes use of induced representations in an essential way.

The Schwinger Representation of a Group

The Schwinger construction for SU (2) briefly described in Chapter 3 has very interesting features contrasting with the regular representations.