As was mentioned at the outset, symmetry as exemplified through group theory brings added information to the solution of physical problems, especially in the application of harmonic analysis. The heart of this information is encapsulated in the so called irreducible representations of the group elements. It should be stated at the outset that the irreducible representations used in most applications are readily available in tabulated form. Yet much of mathematical group theory is devoted to the derivation and properties of irreducible representations. We do not minimize in any way the importance of that material; it is necessary for a clear understanding of the applicability of the mathematical machinery and its physical interpretation. Our objective here is only to touch on a few of the central results used in the applications. Perhaps this may motivate the reader to look further into the subject.

The key property for the application of point groups to physical problems is that for a finite group all representations may be "built up" from a finite number of "distinct" irreducible representations. The number of distinct irreducible representations is equal to the number of classes in the group. Furthermore, the regular representation contains each irregular representation a number of times equal to the number of dimensions of that irreducible representation. Thus, if £a is the dimension of the a-th irreducible representation,

E 4 = |G|, (2.10)

k

where | G| is the order of the group G to be satisfied.

Let us illustrate this with the group C3 that was discussed previously. To identify the classes in C3, as before, we compute a table of TQT, see Table 5. The elements that transform into

Q/T	E	C3	C3
E	E	E	E
C3	C3	C3	C3
C2 C3	C2 c3	C2 c3	C2 c3

Table 5. Classes of Group G3

themselves form a class. There are three classes in C3, denoted as E, C3, and C^ and therefore there are three irreducible representations in the regular representation. The condition

i2+i2 + 4 = з

can only be satisfied by t = £2 = £3 = 1. Therefore, there are three distinct one-dimensional representations. These are the building blocks for decomposing the regular representation to irreducible representations, and can be found in tables:

D(1) (E) = 1 D(1) (С3 ) = 1 D(1) (С3 ) = 1	(2.11)
D(2)(E) = 1 D^O = ш D(2)(С3) = ш*	(2.12)
D(3)(E) = 1 D(3)(C3)= ш* D(3)(С3) = ш,	(2.13)

where ш = exp(2ni/3). The element in each of the three irreducible representation conform to the multiplication of point group C3.

These low dimension irreducible representations are used to build an irreducible representation from the regular representation of the operator Oc3 for example, as follows.

The regular representation has the form of a full matrix,

Ои(Сз) Dii(C3) Оіз(Сз) 010

Оц(Сз) Dii(C3) Оіз(Сз) = 00 1 .

Озі(Сз) D3i(C3) D33C) 100

The irreducible representation has the form of a diagonal (block diagonal in the general case) matrix,

D1^) 0) 0 10 0

0 D2(С3) 0 = 0 ш 0 .

0 0 D3(Q) 0 0 ш*

The mathematical relationship is discussed at length in all texts on the subject, and will not be repeated here. We assume the irreducible representations are known. Of interest is the information for the solution of physical problem, that is associated with irreducible representations.

Recall that starting with an arbitrary function f (r) belonging to a function space L (a Hilbert space for example), we can generate a set of functions fj,…,f|G| that span an invariant subspace Ls C L. This process requires the matrices of coordinate transformations g1,…,g|G| that form the symmetry group G of interest. The diagonal structure of the irreducible representations of G tells us that there exists a set of basis functions {f,f2,…,fn} that split the subspace Ls further into subspaces invariant under the symmetry group G, and are associated with each irreducible representation D(1) (g), D(2) (g),…, D(nc) (g) where nc is the number of classes in G. That is

Ls = Li U L2 U… Lnc (2.14)

and thus an arbitrary function f (r) Є Ls is expressible as a sum of functions that act as basis function in the invariant subspaces associated with each irreducible representation D(a) (g), a = 1,…,nc as

nc

f (r) = E fa (r). (2.15)

a=1

If the decomposition of the regular representation contains irreducible representations of dimension greater than one, we have for each basis function that "belongs to the a-th irreducible representation"

t*

fa (r) = E f»* (r) (2.16)

t=1

where t* is the dimension of the a-th irreducible representation.

The question now remains how do we obtain f * (r), the basis function of each irreducible representation?

To this end we can apply a projection operator that resolves a given function f (r) into basis functions associated with each irreducible representation. This projection operator is defined as

P* = G E Du(8)°g. (2.17)

N geG

The information needed to construct this operator-the coordinate transformations, the irreducible representations-are known in the case of the point groups encountered in practice. So, for example, the i-th basis function of the a irreducible representation that is t* dimensional for a symmetry group with G elements is constructed from an arbitrary function f (r) in invariant space Ls as

fi(r) = G E Dat(g)Ogf (r). (2.18)

|G| geG

This decomposition creates a complete finite set of orthogonal basis functions.

In practice, a more simple projection operator is generally sufficient. This is due to the fact that the Dat (g) ‘s (the diagonal elements of a multidimensional irreducible representation) are quantities that are intrinsic properties of the irreducible representation D* (g). That is they are invariant under the change of coordinates.

Furthermore, the sum of the diagonal elements, or trace, of the irreducible representation Da (g) is also invariant under a change of coordinates. In group theory this trace is denoted by the symbol X* (g) and

X* (g) = E D*t(g), (2.19)

i=l

and referred to as the character of element g e G in the a-th irreducible representation. There are tables of characters for all the point groups of physical interest.

The projection operator in terms of characters is given as

P* = G E X* (g)Og (2.20)

|G| geG

so that the basis functions are

f * (r) = G E X*gOgf (r^

|G| geG

and f (r) is decomposed into a complete finite set of orthogonal functions, with one for each irreducible representation irrespective of its dimension.