χ
R
ð Þ g ν
ð Þ ¼
X
α
q α χ
α
ð Þ g ν
ð Þ,
ð18:96Þ
where χ
(α) is a trace of the irreducible representation D
(α) . Using (18.83) and
(18.94),
q α ¼
1
n
X
g
χ
α
ð Þ g
ð Þ
à χ
R
ð Þ g
ð Þ ¼
1
n
χ
α
ð Þ e
ð Þ
à χ
R
ð Þ e
ð Þ ¼
1
n
χ
α
ð Þ e
ð Þ
à n ¼ χ
α
ð Þ e
ð Þ
Ã
¼ d α :
ð18:97Þ
Note that a dimension d α of the representation D
(α) is equal to a trace of its identity
matrix. Also notice from (18.95) and (18.97) that D
(R) contains every irreducible
representation D
(α) d α times. To show it more clearly, Table 18.4 gives a character
table of C 2v . The regular representation matrices are given in Example 18.2. For this,
we have
D
R
ð Þ C 2v
ð Þ ¼ A 1 þ A 2 þ B 1 þ B 2 :
ð18:98Þ
This relation obviously indicates that all the irreducible representations of C 2v are
contained one time (that is equal to the dimension of representation of C 2v ).
Returning to (18.94) and (18.96) and replacing q α with d α there, we get
X
α
d α χ
α
ð Þ g ν
ð Þ ¼
n
0
8
> <
> :
for g ν ¼ e,
for g ν 6 ¼ e:
ð18:99Þ
In particular, when g ν ¼ e, again we have χ
(α) (e) ¼ d α (d α is a real number!).
That is,
X
α
d α
2
¼ n:
ð18:100Þ
This is very important relation in the representation theory of a finite group in that
(18.100) sets an upper limit to the number of irreducible representations and their
dimensions. That number cannot exceed the order of a group.
Table 18.4 Character table
of C 2v
C 2v
E
C 2 (z)
σ v (zx)
σ
0
v yz
ð Þ
A 1
1
1
1
1
z; x
2
, y
2
, z
2
A 2
1
1
À1
À1
xy
B 1
1
À1
1
À1
x; zx
B 2
1
À1
À1
1
y; yz
18.5 Regular Representation and Group Algebra
703