29.2 CHOOSING AN APPROPRIATE FORMALISM
(a) (b) (c)P P P
R Q R^1 Q R^1 Q
1
3 2 22
3 3
Figure 29.2 Diagram (a) shows the definition of the basis vector, (b) shows
the effect of applying a clockwise rotation of 2π/3 and (c) shows the effect of
applying a reflection in the mirror axis through Q.from which it can be verified thatD(C)D(B)=D(E).
Whilst a representation obtained in this way necessarily has the same dimensionas the order of the group it represents, there are, in general, square matrices of
both smaller and larger dimensions that can be used to represent the group,
though their existence may be less obvious.
One possibility that arises when the group elements are symmetry opera-tions on an object whose position and orientation can be referred to a space
coordinate system is called thenatural representation. In it the representative
matricesD(X) describe, in terms of a fixed coordinate system, what happens
to a coordinate system that moves with the object whenXis applied. There
is usually some redundancy of the coordinates used in this type of represen-
tation, since interparticle distances are fixed and fewer than 3Ncoordinates,
whereNis the number of identical particles, are needed to specify uniquely
the object’s position and orientation. Subsection 29.11.1 gives an example that
illustrates both the advantages and disadvantages of the natural representation.
We continue here with an example of a natural representation that has no such
redundancy.
Use the fact that the group considered in the previous worked example is isomorphic to
the group of two-dimensional symmetry operations on an equilateral triangle to generate a
three-dimensional representation of the group.Label the triangle’s corners as 1, 2, 3 and three fixed points in space as P, Q, R, so that
initially corner 1 lies at point P, 2 lies at point Q, and 3 at point R. We take P, Q, R as
the components of the basis vector.
In figure 29.2, (a) shows the initial configuration and also, formally, the result of applying
the identityIto the triangle; it is therefore described by the basis vector, (P Q R)T.
Diagram (b) shows the the effect of a clockwise rotation by 2π/3, corresponding to
elementAin the previous example; the new column matrix is (Q R P)T.
Diagram (c) shows the effect of a typical mirror reflection – the one that leaves the
corner at point Q unchanged (elementDin table 28.8 and the previous example); the new
column matrix is now (R Q P)T.
In similar fashion it can be concluded that the column matrix corresponding to element
B,rotationby4π/3, is (R P Q)T, and that the other two reflectionsCandEresult in