If K j 6 ¼ K i
0 (i.e., K j is not identical with K i
0 as a set), K i K j does not contain e.
In fact, suppose that for g ρ 2 K i there were a group element b 2 K j such that
bg ρ ¼ e. Then, we would have
b ¼ g ρ
À1 and b 2 K i
0 :
ð18:121Þ
This would mean that b 2 K j
T
K i
0 , implying that K j ¼ K i
0 by definition of
class. It is in contradiction to K j 6 ¼ K i
0 . Consequently, K i K j does not contain e.
Taking the product of the classes K i and K i
0 , we obtain the identity e precisely k i
times. Rewriting (18.117), we have
K i K j ¼ c ij1 K 1 þ
X
l6 ¼1
c ijl K l ,
ð18:122Þ
where K 1 ¼ {e}. As mentioned below, we are most interested in the first term
in (18.122). In (18.122), if K j ¼ K i
0 , we have
c ij1 ¼ k i :
ð18:123Þ
On the other hand, if K j 6 ¼ K i
0 , c ij1 ¼ 0. Summarizing the above arguments,
we get
c ij1 ¼
k i
0
8
> <
> :
for K j ¼ K i
0 ,
for K j 6 ¼ K i
0 :
ð18:124Þ
Or, symbolically we write
c ij1 ¼ k i δ ji
0 :
ð18:125Þ
18.6 Classes and Irreducible Representations
After the aforementioned considerations, we have the following theorem:
Theorem 18.5 Let ℊ ¼ {g 1 , g 2 , Á Á Á, g n } be a group. A trace of irreducible representations satisfies the following orthogonality relation:
X n r
α¼1
χ
α
ð Þ K i
ð Þχ
α
ð Þ K j
À Á Ã ¼
n
k i
δ ij ,
ð18:126Þ
where summation α is taken over all inequivalent n r irreducible representations; K i
and K j indicate conjugacy classes; k i denotes the number of elements contained in the
i-th class K i .
18.6 Classes and Irreducible Representations
707