The application of the information in an abstract group to a physical problem, especially to the calculation of the solution of the boundary value problem that models the physical setting, requires a mathematical "connection" between the two. This connection originates with the transformations of coordinates that define the symmetry operations reflected in the actions of a point group.

As a simple illustration, let us again consider the abstract group G = {E, A, B, a’, b’, c’} in the form of its realization in forms of rotations and reflections of an equilateral triangle, namely the point group C3v = {E, C3, C^, dv, d’v, d"v}. Let this group be consistent with a physical problem in terms of, for example, material distribution and the geometry of the boundary. Furthermore, let us consider a two-dimensional vector space with an orthonormal basis {ei, Є2} relative to which the physical model is defined. Each operation by an element g of the group C3v can be represented by its action on an arbitrary vector r in a two-dimensional vector space. In the usual symbolic form we have

r’ = D(g)r for all g Є C3v

and where r’ = Г1Є1 + Г2Є2 is the transformed vector, and D(g) is the matrix operator associated with the action of group element g Є C3v. It is well known from linear algebra that the matrix representation of operator D (g) for each g Є C3v is obtained by its action on

j=1

and that the transpose of matrix Dji(g) gives the action of the group element g on the coordinates of the vector r as

2

r = E D-j 1(g)Tj, i = 1,2. (2.1)

j=1

For the point group C3v we obtain the following six matrix representations. To spare room we replace the matrices by permutations:

E = (1,2,3); D(C3) = (3,1,2); D(C2) = (2,3,1); (2.2)

D(^1) = (1,3,2); D(^2) = (3,2,1); Dfo) = (2,1,3). (2.3)

These matrices satisfy the group multiplication table of C3v, and therefore also the multiplication table of the abstract group G that is isomorphic to C3v. We note that this is not the only matrix representation of C3v. There are two one-dimensional representations, in particular that also satisfy the multiplication table of C3v, and will be of interest later. These are

D(E) = D(C3) = D(C32) = D(av) = D(^) = D(a"v) = E2, (2.4)

where E2 is the 2 x 2 identity matrix; and

D(E) = D(C3) = D(C2) = E2 D(av) = D(^) = D(a"v) = — E2. (2.5)

The role played by these representations will become clear in later discussions of irreducible representations of groups, and their actions on function spaces.