Characters

REPRESENTATION THEORY
(d) No explicit calculation is needed to see that if i = j = k = l = 1, with
(λ)
(µ)
D̂ = D̂ = A1 (or A2 ), then each term in the sum is either 12 or (−1)2
and the total is 6, as predicted by the right-hand side of (29.13) since g = 6
and nλ = 1.
29.6 Characters
The actual matrices of general representations and irreps are cumbersome to
work with, and they are not unique since there is always the freedom to change
the coordinate system, i.e. the components of the basis vector (see section 29.3),
and hence the entries in the matrices. However, one thing that does not change
for a matrix under such an equivalence (similarity) transformation – i.e. under
a change of basis – is the trace of the matrix. This was shown in chapter 8,
but is repeated here. The trace of a matrix A is the sum of its diagonal elements,
n
Aii
Tr A =
i=1
or, using the summation convention (section 26.1), simply Aii . Under a similarity
transformation, again using the summation convention,
[DQ (X)]ii = [Q−1 ]ij [D(X)]jk [Q]ki
= [D(X)]jk [Q]ki [Q−1 ]ij
= [D(X)]jk [I]kj
= [D(X)]jj ,
showing that the traces of equivalent matrices are equal.
This fact can be used to greatly simplify work with representations, though with
some partial loss of the information content of the full matrices. For example,
using trace values alone it is not possible to distinguish between the two groups
known as 4mm and 4̄2m, or as C4v and D2d respectively, even though the two
groups are not isomorphic. To make use of these simplifications we now define
the characters of a representation.
Definition. The characters χ(D) of a representation D of a group G are defined as
the traces of the matrices D(X), one for each element X of G.
At this stage there will be g characters, but, as we noted in subsection 28.7.3,
elements A, B of G in the same conjugacy class are connected by equations of
the form B = X −1 AX. It follows that their matrix representations are connected
by corresponding equations of the form D(B) = D(X −1 )D(A)D(X), and so by the
argument just given their representations will have equal traces and hence equal
characters. Thus elements in the same conjugacy class have the same characters,
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29.6 CHARACTERS
3m
I
A, B
C, D, E
A1
A2
E
1
1
2
1
1
−1
1
−1
0
z; z 2 ; x2 + y 2
Rz
(x, y); (xz, yz); (Rx , Ry ); (x2 − y 2 , 2xy)
Table 29.1 The character table for the irreps of group 3m (C3v or S3 ). The
right-hand column lists some common functions that transform according to
the irrep against which each is shown (see text).
though, in general, these will vary from one representation to another. However,
it might also happen that two or more conjugacy classes have the same characters
in a representation – indeed, in the trivial irrep A1 , see (29.12), every element
inevitably has the character 1.
For the irrep A2 of the group 3m, the classes {I}, {A, B} and {C, D, E} have
characters 1, 1 and −1, respectively, whilst they have characters 2, −1 and 0
respectively in irrep E.
We are thus able to draw up a character table for the group 3m as shown
in table 29.1. This table holds in compact form most of the important information on the behaviour of functions under the two-dimensional rotational and
reflection symmetries of an equilateral triangle, i.e. under the elements of group
3m. The entry under I for any irrep gives the dimension of the irrep, since it
is equal to the trace of the unit matrix whose dimension is equal to that of
the irrep. In other words, for the λth irrep χ(λ) (I) = nλ , where nλ is its dimension.
In the extreme right-hand column we list some common functions of Cartesian
coordinates that transform, under the group 3m, according to the irrep on whose
line they are listed. Thus, as we have seen, z, z 2 , and x2 + y 2 are all unchanged
by the group operations (though x and y individually are affected) and so are
listed against the one-dimensional irrep A1 . Each of the pairs (x, y), (xz, yz), and
(x2 − y 2 , 2xy), however, is mixed as a pair by some of the operations, and so these
pairs are listed against the two-dimensional irrep E: each pair forms a basis set
for this irrep.
The quantities Rx , Ry and Rz refer to rotations about the indicated axes;
they transform in the same way as the corresponding components of angular
momentum J, and their behaviour can be established by examining how the
components of J = r × p transform under the operations of the group. To do
this explicitly is beyond the scope of this book. However, it can be noted that
Rz , being listed opposite the one-dimensional A2 , is unchanged by I and by the
rotations A and B but changes sign under the mirror reflections C, D, and E, as
would be expected.
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REPRESENTATION THEORY
29.6.1 Orthogonality property of characters
Some of the most important properties of characters can be deduced from the
orthogonality theorem (29.13),
∗ (µ)
(λ)
g
D̂ (X)
D̂ (X) = δik δjl δλµ .
ij
kl
n
λ
X
If we set j = i and l = k, so that both factors in any particular term in the
summation refer to diagonal elements of the representative matrices, and then
sum both sides over i and k, we obtain
nµ nλ X
∗ (µ)
(λ)
D̂ (X)
D̂ (X)
ii
i=1 k=1
kk
=
nµ
nλ g δik δik δλµ .
nλ
i=1 k=1
Expressed in term of characters, this reads
nλ
nλ
∗
g g χ(λ) (X) χ(µ) (X) =
δii2 δλµ =
1 × δλµ = gδλµ .
nλ
nλ
X
i=1
i=1
(29.14)
In words, the (g-component) ‘vectors’ formed from the characters of the various
irreps of a group are mutually orthogonal, but each one has a squared magnitude
(the sum of the squares of its components) equal to the order of the group.
Since, as noted in the previous subsection, group elements in the same class
have the same characters, (29.14) can be written as a sum over classes rather than
elements. If ci denotes the number of elements in class Ci and Xi any element of
Ci , then
∗
ci χ(λ) (Xi ) χ(µ) (Xi ) = gδλµ .
(29.15)
i
Although we do not prove it here, there also exists a ‘completeness’ relation for
characters. It makes a statement about the products of characters for a fixed pair
of group elements, X1 and X2 , when the products are summed over all possible
irreps of the group. This is the converse of the summation process defined by
(29.14). The completeness relation states that
∗
g
χ(λ) (X1 ) χ(λ) (X2 ) = δC1 C2 ,
(29.16)
c1
λ
where element X1 belongs to conjugacy class C1 and X2 belongs to C2 . Thus the
sum is zero unless X1 and X2 belong to the same class. For table 29.1 we can
verify that these results are valid.
(λ)
(i) For D̂
= D̂
(µ)
= A1 or A2 , (29.15) reads
1(1) + 2(1) + 3(1) = 6,
1094