Choosing an appropriate formalism

REPRESENTATION THEORY
Finally, for ozone, which is angular rather than linear, symmetry does not
place such tight constraints. A dipole-moment component parallel to the axis
BB (figure 29.1(c)) is possible, since there is no symmetry operation that reverses
the component in that direction and at the same time carries the molecule into
an indistinguishable copy of itself. However, a dipole moment perpendicular to
BB is not possible, since a rotation of π about BB would both reverse any
such component and carry the ozone molecule into itself – two contradictory
conclusions unless the component is zero.
In summary, symmetry requirements appear in the form that some or all
components of permanent electric dipoles in molecules are forbidden; they do
not show that the other components do exist, only that they may. The greater
the symmetry of the molecule, the tighter the restrictions on potentially non-zero
components of its dipole moment.
In section 23.11 other, more complicated, physical situations will be analysed
using results derived from representation theory. In anticipation of these results,
and since it may help the reader to understand where the developments in the
next nine sections are leading, we make here a broad, powerful, but rather formal,
statement as follows.
If a physical system is such that after the application of particular rotations or
reflections (or a combination of the two) the final system is indistinguishable from
the original system then its behaviour, and hence the functions that describe its
behaviour, must have the corresponding property of invariance when subjected to
the same rotations and reflections.
29.2 Choosing an appropriate formalism
As mentioned in the introduction to this chapter, the elements of a finite group
G can be represented by matrices; this is done in the following way. A suitable
column matrix u, known as a basis vector,§ is chosen and is written in terms of
its components ui , the basis functions, as u = (u1 u2 · · · un )T . The ui may be of
a variety of natures, e.g. numbers, coordinates, functions or even a set of labels,
though for any one basis vector they will all be of the same kind.
Once chosen, the basis vector can be used to generate an n-dimensional representation of the group as follows. An element X of the group is selected and
its effect on each basis function ui is determined. If the action of X on u1 is to
produce u1 , etc. then the set of equations
ui = Xui
§
(29.1)
This usage of the term basis vector is not exactly the same as that introduced in subsection 8.1.1.
1078
29.2 CHOOSING AN APPROPRIATE FORMALISM
generates a new column matrix u = (u1 u2 · · · un )T . Having established u and u
we can determine the n × n matrix, M(X) say, that connects them by
u = M(X)u.
(29.2)
It may seem natural to use the matrix M(X) so generated as the representative
matrix of the element X; in fact, because we have already chosen the convention
whereby Z = XY implies that the effect of applying element Z is the same as that
of first applying Y and then applying X to the result, one further step has to be
taken. So that the representative matrices D(X) may follow the same convention,
i.e.
D(Z) = D(X)D(Y ),
and at the same time respect the normal rules of matrix multiplication, it is
necessary to take the transpose of M(X) as the representative matrix D(X).
Explicitly,
D(X) = MT (X)
(29.3)
u = DT (X)u.
(29.4)
and (29.2) becomes
Thus the procedure for determining the matrix D(X) that represents the group
element X in a representation based on basis vector u is summarised by equations
(29.1)–(29.4).§
This procedure is then repeated for each element X of the group, and the
resulting set of n × n matrices D = {D(X)} is said to be the n-dimensional
representation of G having u as its basis. The need to take the transpose of each
matrix M(X) is not of any fundamental significance, since the only thing that
really matters is whether the matrices D(X) have the appropriate multiplication
properties – and, as defined, they do.
In cases in which the basis functions are labels, the actions of the group
elements are such as to cause rearrangements of the labels. Correspondingly the
matrices D(X) contain only ‘1’s and ‘0’s as entries; each row and each column
contains a single ‘1’.
§
An alternative procedure in which a row vector is used as the basis vector is possible. Defining
equations of the form uT X = uT D(X) are used, and no additional transpositions are needed to
define the representative matrices. However, row-matrix equations are cumbersome to write out
and in all other parts of this book we have adopted the convention of writing operators (here the
group element) to the left of the object on which they operate (here the basis vector).
1079
REPRESENTATION THEORY
For the group S3 of permutations on three objects, which has group multiplication table 28.8 on p. 1055, with (in cycle notation)
I = (1)(2)(3),
C = (1)(2 3),
A = (1 2 3),
D = (3)(1 2),
B = (1 3 2
E = (2)(1 3),
use as the components of a basis vector the ordered letter triplets
u1 = {P Q R},
u4 = {P R Q},
u2 = {Q R P},
u5 = {Q P R},
u3 = {R P Q},
u6 = {R Q P}.
Generate a six-dimensional representation D = {D(X)} of the group and confirm that the
representative matrices multiply according to table 28.8, e.g.
D(C)D(B) = D(E).
It is immediate that the identity permutation I = (1)(2)(3) leaves all ui unchanged, i.e.
ui = ui for all i. The representative matrix D(I) is thus I6 , the 6 × 6 unit matrix.
We next take X as the permutation A = (1 2 3) and, using (29.1), let it act on each of
the components of the basis vector:
u1 = Au1 = (1 2 3){P Q R} = {Q R P} = u2
u2 = Au2 = (1 2 3){Q R P} = {R P Q} = u3
..
..
.
.
u6 = Au6 = (1 2 3){R Q P} = {Q P R} = u5 .
The matrix M(A) has to be such that u = M(A)u (here dots replace zeros to aid readability):

 


u1
· 1 · · · ·
u2
 u3   · · 1 · · ·   u2 

 


 u   1 · · · · ·   u3 
u =  1  = 


 ≡ M(A)u.
 u6   · · · · · 1   u4 
 u4   · · · 1 · ·   u5 
· · · · 1 ·
u5
u6
D(A) is then equal to MT (A).
The other D(X) are determined in a similar way. In general, if
Xui = uj ,
then [M(X)]ij = 1, leading to [D(X)]ji = 1 and [D(X)]jk = 0 for k = i. For example,
Cu3 = (1)(23){R P Q} = {R Q P} = u6
implies that [D(C)]63

·
 ·

 ·
D(C) = 
 1
 ·
·
= 1 and [D(C)]6k = 0 for k = 1, 2, 4, 5, 6. When calculated in full



· 1 · · · ·
· · 1 · ·
 · · 1 · · · 
· · · 1 · 



· · · · 1 
 1 · · · · · 
,
D(B) = 
,

· · · · · 
 · · · · · 1 
 · · · 1 · · 
1 · · · · 
· · · · 1 ·
· 1 · · ·


· · · · · 1
 · · · 1 · · 


 · · · · 1 · 
D(E) = 
,
·
1
·
·
·
·


 · · 1 · · · 
1 · · · · ·
1080
29.2 CHOOSING AN APPROPRIATE FORMALISM
R
P
P
P
1
3
3
2
3
(a)
Q
R
2
1
Q
(b)
R
2 Q
1
(c)
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 that D(C)D(B) = D(E). Whilst a representation obtained in this way necessarily has the same dimension
as 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 operations on an object whose position and orientation can be referred to a space
coordinate system is called the natural representation. In it the representative
matrices D(X) describe, in terms of a fixed coordinate system, what happens
to a coordinate system that moves with the object when X is applied. There
is usually some redundancy of the coordinates used in this type of representation, since interparticle distances are fixed and fewer than 3N coordinates,
where N is 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 identity I to 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
element A in 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 (element D in 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, rotation by 4π/3, is (R P Q)T , and that the other two reflections C and E result in
1081
REPRESENTATION THEORY
column matrices (P R Q)T and (Q P R)T respectively. The forms of the representative
matrices Mnat (X), (29.2), are now determined by equations such as, for element E,



 
P
0 1 0
Q
 P  =  1 0 0  Q 
0 0 1
R
R
implying that

0
D (E) =  1
0
nat
1
0
0
T 
0
0
0  = 1
1
0
In this way the complete representation is obtained



1 0 0
0 0
nat
nat


0 1 0
D (I) =
, D (A) =  1 0
0 0 1
0 1



1 0 0
0 0
Dnat (C) =  0 0 1  , Dnat (D) =  0 1
0 1 0
1 0
1
0
0

0
0 .
1
as

1
0 ,
0

1
0 ,
0

0
D (B) =  0
1

0
Dnat (E) =  1
0
nat
1
0
0
1
0
0

0
1 ,
0

0
0 .
1
It should be emphasised that although the group contains six elements this representation
is three-dimensional. We will concentrate on matrix representations of finite groups, particularly
rotation and reflection groups (the so-called crystal point groups). The general
ideas carry over to infinite groups, such as the continuous rotation groups, but in
a book such as this, which aims to cover many areas of applicable mathematics,
some topics can only be mentioned and not explored. We now give the formal
definition of a representation.
Definition. A representation D = {D(X)} of a group G is an assignment of a nonsingular square n × n matrix D(X) to each element X belonging to G, such that
(i) D(I) = In , the unit n × n matrix,
(ii) D(X)D(Y ) = D(XY ) for any two elements X and Y belonging to G, i.e. the
matrices multiply in the same way as the group elements they represent.
As mentioned previously, a representation by n × n matrices is said to be an
n-dimensional representation of G. The dimension n is not to be confused with
g, the order of the group, which gives the number of matrices needed in the
representation, though they might not all be different.
A consequence of the two defining conditions for a representation is that the
matrix associated with the inverse of X is the inverse of the matrix associated
with X. This follows immediately from setting Y = X −1 in (ii):
D(X)D(X −1 ) = D(XX −1 ) = D(I) = In ;
hence
D(X −1 ) = [D(X)]−1 .
1082
29.2 CHOOSING AN APPROPRIATE FORMALISM
As an example, the four-element Abelian group that consists of the set {1, i, −1, −i}
under ordinary multiplication has a two-dimensional representation based on the
column matrix (1 i)T :
1 0
0 −1
D(1) =
,
D(i) =
,
0 1
1 0
−1 0
0 1
D(−1) =
, D(−i) =
.
0 −1
−1 0
The reader should check that D(i)D(−i) = D(1), D(i)D(i) = D(−1) etc., i.e. that
the matrices do have exactly the same multiplication properties as the elements
of the group. Having done so, the reader may also wonder why anybody would
bother with the representative matrices, when the original elements are so much
simpler to handle! As we will see later, once some general properties of matrix
representations have been established, the analysis of large groups, both Abelian
and non-Abelian, can be reduced to routine, almost cookbook, procedures.
An n-dimensional representation of G is a homomorphism of G into the set of
invertible n × n matrices (i.e. n × n matrices that have inverses or, equivalently,
have non-zero determinants); this set is usually known as the general linear group
and denoted by GL(n). In general the same matrix may represent more than one
element of G; if, however, all the matrices representing the elements of G are
different then the representation is said to be faithful, and the homomorphism
becomes an isomorphism onto a subgroup of GL(n).
A trivial but important representation is D(X) = In for all elements X of G.
Clearly both of the defining relationships are satisfied, and there is no restriction
on the value of n. However, such a representation is not a faithful one.
To sum up, in the context of a rotation–reflection group, the transposes of
the set of n × n matrices D(X) that make up a representation D may be thought
of as describing what happens to an n-component basis vector of coordinates,
(x y · · · )T , or of functions, (Ψ1 Ψ2 · · · )T , the Ψi themselves being functions
of coordinates, when the group operation X is carried out on each of the
coordinates or functions. For example, to return to the symmetry operations
on an equilateral triangle, the clockwise rotation by 2π/3, R, carries the threedimensional basis vector (x y z)T into the column matrix


√
− 21 x + 23 y
 √

 − 3x − 1y 


2
2
z
whilst the two-dimensional basis vector of functions (r 2 3z 2 − r 2 )T is unaltered,
as neither r nor z is changed by the rotation. The fact that z is unchanged by
any of the operations of the group shows that the components x, y, z actually
divide (i.e. are ‘reducible’, to anticipate a more formal description) into two sets:
1083