Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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386 10 Group Theory
We may again assume that both D
1
and D
2
are unitary and that n
1
n
2
.Usingthe
fact that the representations are both unitary and irreducible, the hermitian adjoint takes
the form
M
D
1
g
D
2
g
M
M
D
1
g
1
D
2
g
1
M
(10.66)
Multiplying both sides by M and using the stipulated commutation property, we find
MM
D
1
g
1
MD
2
g
1
M
MM
D
1
g
1
D
1
g
1
MM
(10.67)
Therefore, according to Schur’s lemma, MM
must be a multiple of the unit matrix. First
suppose that n
1
n
2
such that M is a square matrix and MM
cI where c is a number.
Then
Det
MM
DetM DetM
DetM
2
c (10.68)
requires that c be nonnegative. If M 0, then DetM0 and M is invertible. Next
suppose that n
1
<n
2
and consider the n
2
n
2
matrix N formed by appending n
2
n
1
columns of zeros to M. Matrix multiplication
NN
i, j
n
2
k1
N
i,k
N
j,k
n
1
k1
M
i,k
M
j,k
MM
i, j
(10.69)
shows that NN
MM
because the appended columns do not contribute. However,
DetN0 by construction, such that DetM
2
DetN
2
0 c 0. There-
fore, if n
1
n
2
, then M 0.
Orthogonality Theorem
We are finally ready to derive the orthogonality theorem for irreducible representations.
Let D
i
and D
j
represent irreducible representations of group G of order N
G
in terms
of nonsingular matrices of dimension n
i
and n
j
that are either identical or inequivalent;
equivalent representations of the same dimensions are excluded. Then
gG
D
i
Μ,Ν
g
D
j
Μ
'
,Ν
'
g
N
G
n
i
∆
i, j
∆
Μ,Μ
'
∆
Ν,Ν
'
(10.70)
where the summation runs over all group elements. This theorem will play a central role in
determining the number of inequivalent irreducible representations that exist with specified
dimensions.
Consider
M
gG
D
j
gXD
i
g
1
(10.71)
10.3 Representations 387
where X is an arbitrary matrix and both M and X have dimensions n
j
n
i
. We can demon-
strate that M satisfies the condition of the third lemma as follows.
D
j
gM
hG
D
j
ghXD
i
h
1
hG
D
j
ghXD
i
h
1
g
1
g
aG
D
j
aXD
i
a
1
D
i
g
MD
i
g
(10.72)
Any particular matrix element takes the form
M
ΑΜ
gG
Β
'
,Γ
D
j
Α,Β
'
gX
Β
'
,Γ
D
i
Γ,Μ
g
1
(10.73)
First suppose that n
i
n
j
M 0. Recognizing that the matrix elements X
Β
'
,Γ
are
arbitrary, we are free to choose X
Β
'
,Γ
∆
Β,Β
'
∆
Γ,Ν
, such that
0
gG
D
j
Α,Β
gD
i
Ν,Μ
g
1
(10.74)
The unitarity of D
1
then requires
0
gG
D
i
Μ,Ν
g
D
j
Α,Β
g (10.75)
for n
i
n
j
.
Next suppose that n
i
n
j
, such that either M 0orD
i
G D
j
. The former shows
that inequivalent representations are orthogonal. Different equivalent representations are
excluded by the conditions of the theorem, leaving us only to deduce the normalization of
the sum over products of matrix elements for the same representation. Again using the fact
that M is a multiple of the unit matrix gives
gG
Γ,Κ
D
i
Μ,Γ
g
D
i
Α,Κ
gX
Κ,Γ
c∆
Α,Μ
(10.76)
for any X. Making an inspired choice
X
Κ,Γ
∆
Κ,Β
∆
Γ,Ν
gG
D
i
Μ,Ν
g
D
i
Α,Β
gc∆
Α,Μ
(10.77)
we can use the unitarity of D
i
to perform the sum over the n
i
identical diagonal matrix
elements to obtain
Μ
gG
D
i
Μ,Ν
g
D
i
Μ,Β
gcn
i
N
G
∆
Β,Ν
c
N
G
n
i
∆
Β,Ν
(10.78)
388 10 Group Theory
Therefore, we finally obtain the orthogonality theorem
gG
D
i
Μ,Ν
g
D
j
Α,Β
g
N
G
n
i
∆
Α,Μ
∆
Β,Ν
∆
i, j
(10.79)
An important special case is that the normalization of matrix elements
gG
D
i
Μ,Ν
g
2
N
G
n
i
(10.80)
summed over group elements is independent of the indices of the matrices.
10.3.4 Character
Simple Characters
In the face of an unlimited number of equivalent irreducible representations that can be
produced by similarity transformations, it is desirable to describe irreps in terms of invari-
ant properties. An obvious candidate is the trace of the matrices, which is invariant under
similarity transformations. Furthermore, because all members of a class are related by
similarity transformations, the trace of representation matrices must be the same for every
member of a class. Therefore, the character Χ
j
k
of class
k
in representation D
j
is
defined as Tr
D
j
g
where g
k
. The characters for irreducible representations are
described as simple while those for reducible representations are composite. In this sec-
tion we use the orthogonality theorem to develop several valuable relationships for simple
characters and in the next we consider composite characters.
Specializing the orthogonality theorem for inequivalent irreducible representations to
diagonal elements
gG
D
i
Μ,Μ
g
D
j
Α,Α
g
N
G
n
i
∆
i, j
∆
Α,Μ
(10.81)
and summing over indices, we find
gG
Χ
i
g
Χ
j
gN
G
∆
i, j
(10.82)
or
k
N
k
Χ
i
k
Χ
j
k
N
G
∆
i, j
(10.83)
where N
k
is the number of elements in class
k
. Evidently, simple characters also satisfy
an orthogonality theorem. When the two irreps are the same, we obtain a formula
g
Χ
i
g
2
k
N
k
Χ
i
k
2
N
G
(10.84)
that severely constrains the total number of inequivalent irreps that exist for a finite group.
10.3 Representations 389
For example, we already know that S
3
contains three classes: I, R, R
2
, and P, PR, P,
R
2
. The character table for S
3
is given in Table 10.4. Each column is labeled by class
and the number of elements in that class. The rows list the characters for representative
matrices in each irreducible representation. Because the identity element is in its own
class, the first column also gives the dimensionality of each irrep. The first row is the
trivial irrep in which each element is represented by the integer 1. For S
3
there is another
one-dimensional representation in terms of parity. These rows are orthogonal, weighted by
class size, and satisfy the normalization condition. No other one-dimensional irreps can
exist. Using the orthogonality conditions, Eq. (10.83), a two-dimensional representation
must satisfy the equations
2 2Χ
3
2
3Χ
3
3
0 (10.85)
2 2Χ
3
2
3Χ
3
3
0 (10.86)
Thus, we find that there is one two-dimensional irrep and no higher-dimensional irreps
are possible because the system of orthogonality equations would be overdetermined. We
also observe that these results are consistent with the normalization condition. Therefore,
these simple considerations are sufficient to construct the character table without actually
producing representation matrices. This exercise suggests that the number of irreducible
representations is equal to the number of classes. This is a general result, but we will forgo
the formal proof.
Table 10.4. Character table for S
3
.
ΧS
3
1
2
2
3
3
D
1
11 1
D
2
11 1
D
3
2 10
Suppose that we define a square matrix
M
i, j
Χ
j
i
(10.87)
from the body of the character table. The inverse of this matrix is found using the orthog-
onality formula
M
1
i, j
N
j
N
G
Χ
i
j
M
1
M
i, j
k
N
k
N
G
Χ
i
k
Χ
j
k
∆
i, j
(10.88)
where the summation ranges over classes. However, because a matrix commutes with its
inverse, we can also express orthogonality in the form
MM
1
i, j
k
N
j
N
G
Χ
k
i
Χ
k
j
∆
i, j
(10.89)
where the summation ranges over irreps.
390 10 Group Theory
Therefore, orthogonality of character
i
N
i
Χ
Μ
i
Χ
Ν
i
N
G
∆
Μ,Ν
(10.90)
Μ
Χ
Μ
i
Χ
Μ
j
N
G
N
i
∆
i, j
(10.91)
can be expressed in terms of either class or irrep and both can be useful in construction of
character tables. Here we chose Greek indices for irreps and Latin indices for class.
Composite Character
Recall that a reducible representation can be cast in block diagonal form with a suitable
similarity transformation. Upon reducing such a representation as fully as possible, such
that each block is irreducible, we write
Dg
Μ
a
Μ
D
Μ
g (10.92)
where the summation ranges over irreducible representations and the nonnegative integers
a
Μ
count the number of times each irrep D
Μ
appears in D. The same irrep may appear
in several equivalent forms within D and a
Μ
counts each equivalent form equally. The
summation is schematic – it enumerates the structure along the diagonal of the block diag-
onalized D. The trace of D then becomes
Χg
Μ
a
Μ
Χ
Μ
gΧ
i
Μ
a
Μ
Χ
Μ
i
(10.93)
where
i
is the class that contains g. Thus, the character for each class in a reducible
representation is the sum over the simple characters for irreducible representations of the
same class weighted by the number of appearances of equivalent forms of those irreps.
The orthogonality theorems for simple characters facilitate projection of a
Μ
i
N
i
Χ
Ν
i
Χ
i
i
Μ
a
Μ
N
i
Χ
Ν
i
Χ
Μ
i
N
G
a
Ν
(10.94)
whereby
a
Μ
1
N
G
i
N
i
Χ
Μ
i
Χ
i
1
N
G
gG
Χ
Μ
g
Χg (10.95)
Suppose that D D
Ν
is irreducible, such that a
Μ
∆
Μ,Ν
. Then
i
N
i
Χ
Μ
i
2
N
G
(10.96)
is consistent with previous results for the character normalization summed over classes of
a particular irrep. In fact, this condition provides a test for irreducibility. For an arbitrary
10.3 Representations 391
representation, one finds
i
N
i
Χ
i
2
2 N
G
(10.97)
with equality if and only if Χ is simple and D is irreducible. Expanding Χ again gives
i
N
i
Μ,Ν
a
Μ
a
Ν
Χ
Μ
i
Χ
Ν
i
N
G
Μ
a
Μ
2
(10.98)
such that
1
N
G
i
N
i
Χ
i
2
Μ
a
Μ
2
2 1 (10.99)
If computation of the quantity on the left gives 1, then the representation is irreducible.
If it happens to be 2, then D contains two different irreps, with each occurring just once,
because the a
Μ
are nonnegative integers. If it happens to be 4, then D contains either one
irrep occurring twice or four different irreps all occurring once.
Dimensionality Theorem
Table 10.5. Multiplication table for S
3
with elements arranged to place I on the main diagonal.
S
3
I abcde
I I abcde
a
1
bI adec
b
1
abI ecd
c
1
cdeIab
d
1
decbIa
e
1
ecdabI
The regular representation of any finite group is constructed by arranging the group
elements to place the identity element along the main diagonal of the group multiplication
table. The matrices D
reg
g are then obtained by replacing g in the multiplication table
by 1 and all other elements by 0. Thus, if we label columns by g
i
we can label rows by g
1
i
.
We illustrate this construction using S
3
again. Using the multiplication table in the form
shown in Table 10.5, we obtain the matrices
392 10 Group Theory
DI
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
100000
010000
001000
000100
000010
000001
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
, (10.100)
Da
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
010000
001000
100000
000010
000001
000100
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
, (10.101)
Db
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
001000
100000
010000
000001
000100
000010
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
, (10.102)
Dc
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
000100
000001
000010
100000
001000
010000
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
, (10.103)
Dd
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
000010
000100
000001
010000
100000
001000
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
, (10.104)
De
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
000001
000010
000100
001000
010000
100000
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
(10.105)
and can easily verify that these obey the group multiplication table.
The character of the identity element of the regular representation is N
G
while the
characters of all other classes are zero by construction. Furthermore, the simple character
Χ
Μ
IN
Μ
for the identity element of an irreducible representation is just the dimension-
ality of irrep Μ. Application of the projection theorem, Eq. (10.95),
10.3 Representations 393
a
reg
Μ
1
N
G
i
N
i
Χ
Μ
i
Χ
reg
i
1
N
G
N
Μ
N
G
a
reg
Μ
N
Μ
(10.106)
then shows that the number of times each irrep appears in the regular representation is
equal to its dimensionality. Finally, the composite character of the identity element of the
regular representation
Χ
reg
I
Μ
a
Μ
Χ
Μ
IN
G
Μ
N
2
Μ
(10.107)
provides a simple relationship between the dimensionality of irreps and the order of the
group. This result is general because neither of these quantities depends upon the repre-
sentation; the fact that we obtained it with the aid of the regular representation does not
limit its generality. Therefore, we obtain the dimensionality theorem
Μ
N
2
Μ
N
G
(10.108)
For example, the fact that N
G
6forS
3
is sufficient to deduce that S
3
supports exactly
two different one-dimensional irreps and one two-dimensional irrep because 1
2
1
2
2
2
6 is the only solution to the dimensionality equation in terms of positive integers N
Μ
.
Furthermore, the regular representation contains both one-dimensional irreps once and the
two-dimensional irrep twice.
10.3.5 Example: Character table for symmetries of a square
In the preceding few subsections we derived many abstract theorems and we need a non-
trivial example to illustrate their use. Consider the symmetries of a square, excluding oper-
ations that twist its sides. These symmetries are among the permutations of the four ver-
tices and can be labeled as
I 1234 (10.109)
R 2341 (10.110)
R
2
3412 (10.111)
R
3
4123 (10.112)
P 4321 (10.113)
PR 1432 (10.114)
PR
2
2143 (10.115)
PR
3
3214 (10.116)
where R indicates a rotation (cyclic permutation) and P a reflection across the x-axis (parity
operation). The combinations of reflection and rotations then fill out the list of permissible
permutations. Table 10.6 provides the group multiplication table with an extra column for
the order of each element. This table shows that multiplication is associative, closed, and
394 10 Group Theory
Table 10.6. Group multiplication table for symmetries of a square.
Square
IRR
2
R
3
PPRPR
2
PR
3
Order
I IRR
2
R
3
PPRPR
2
PR
3
1
R
RR
2
R
3
IPR
3
PPRPR
2
4
R
2
R
2
R
3
IRPR
2
PR
3
PPR2
R
3
R
3
IRR
2
PR PR
2
PR
3
P 4
P
PPRPR
2
PR
3
IRR
2
R
3
2
PR
PR PR
2
PR
3
PR
3
IPRR
2
2
PR
2
PR
2
PR
3
PPRR
2
R
3
IR2
PR
3
PR
3
PPRPR
2
RR
2
R
3
I 2
that there is an inverse for each element. Therefore, this set of operations forms a group of
order 8 and, because each element is a member of S
4
, we conclude that the symmetries of
a square are a subgroup of S
4
.
The calculations below establish class membership.
PRP
1
PRP PPR
3
R
3
R G R
3
(10.117)
RPR
1
PR
3
R
3
PR
2
P G PR
2
(10.118)
RPRR
1
PR
3
PR G PR
3
(10.119)
Thus, the five classes consist of
1
I (10.120)
2
R, R
3
(10.121)
3
R
2
(10.122)
4
P, PR
2
(10.123)
5
PR, PR
3
(10.124)
and there are also five irreps with dimensions d 1, 1, 1, 1, 2, the only solution to d,d 8
among positive integers. Placing the trivial one-dimensional representation D
1
gI on
the first row and the dimensions d on the first column gives us the start of a character
table (10.3.5). We can also use the parity of the permutations as a second one-dimensional
representation giving row 2. By inspection we verify that these rows are orthogonal and
normalized properly when weighted by class size.
Table 10.7. First steps in construction of the character table for symmetries of a square.
Χ
1
2
2
3
2
4
2
5
D
1
11 11 1
D
2
11 11 1
D
3
1
D
4
1
D
5
2
Next, we identify A I,R
2
as an invariant subgroup by examining the multiplication
table and observing that A commutes with all other elements of G. Thus, its factor group is
10.3 Representations 395
F G/A A, RA, PA, PRA and an irrep of F must be an irrep of G by homomorphism.
The multiplication table for F is given in Table 10.8.
Table 10.8.
F
A RA PA PRA
A A RA PA PRA
RA
RA A PRA PA
PA
PA PRA A RA
PRA
PRA PA RA A
Since F is abelian each element is in its own class, such that F supports four one-
dimensional irreps. Every element other than the identity (A) is order 2, such that irreps
must assign them values of 1. Thus, the irreps are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
and 1, 1, 1, 1; it is a simple exercise to verify that these representations obey the mul-
tiplication table for F. We can now use the homomorphism with G to complete rows three
and four of its character table (Tab. 10.3.5). It is comforting to observe that the orthonor-
mality conditions are satisfied at this stage.
Table 10.9.
Χ
1
2
2
3
2
4
2
5
D
1
11 11 1
D
2
11 11 1
D
3
1 111 1
D
4
1 1111
D
5
2
Finally, we use orthogonality relations to produce a system of four linear equations for
the four remaining elements of this table.
2 2Χ
5
2
Χ
5
3
2Χ
5
4
2Χ
5
5
0 (10.125)
2 2Χ
5
2
Χ
5
3
2Χ
5
4
2Χ
5
5
0 (10.126)
2 2Χ
5
2
Χ
5
3
2Χ
5
4
2Χ
5
5
0 (10.127)
2 2Χ
5
2
Χ
5
3
2Χ
5
4
2Χ
5
5
0 (10.128)
After some straightforward algebra, we obtain row five and complete the character table,
as shown in Table 10.10. You should verify, for practice, that both the row and column
orthonormality relations are satisfied for this character table. The reasoning employed here
is typical of that used to construct character tables. There are variations in the sequence
of steps, of course, depending upon what information is available. Sometimes one can
dispense with construction of factor groups; other times more than one is both available
and helpful.