k i k j χ
α
ð Þ K i
ð Þχ
α
ð Þ K j
À Á ¼ d α
X
l
c ijl k l χ
α
ð Þ K l
ð Þ:
ð18:134Þ
Returning to (18.99) and rewriting it, we have
X
α
d α χ
α
ð Þ K i
ð Þ ¼ nδ i1 ,
ð18:135Þ
where again we have K 1 ¼ {e}. With respect to (18.134), we sum over all the
irreducible representations α. Then we have
X
α
k i k j χ
α
ð Þ K i
ð Þχ
α
ð Þ K j
À Á ¼
X
l
c ijl k l
X
α
d α χ
α
ð Þ K l
ð Þ
h
i
¼
X
l
c ijl k l nδ l1 ¼ c ij1 n:
ð18:136Þ
In (18.136) we remark that k 1 ¼ 1, meaning that the number of group elements
that K 1 ¼ {e} contains is 1. Rewriting (18.136), we get
k i k j
X
α
χ
α
ð Þ K i
ð Þχ
α
ð Þ K j
À Á ¼ c ij1 n ¼ k i nδ ji
0 ,
ð18:137Þ
where we used (18.125) with the last equality. Moreover, using
χ
α
ð Þ K i
0
ð Þ ¼ χ
α
ð Þ K i
ð Þ
à ,
ð18:138Þ
we get
k i k j
X
α
χ
α
ð Þ K i
ð Þχ
α
ð Þ K j
À Á Ã ¼ c ij1 n ¼ k i nδ ji :
ð18:139Þ
Rewriting (18.139), we finally get
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
Equations (18.78) and (18.126) are well known as orthogonality relations. So far,
we have no idea about the mutual relationship between the two numbers n c and n r in
magnitude. In (18.78), let us consider a following set S
S ¼
ffiffiffiffi ffi
k 1
p χ
α
ð Þ K 1
ð Þ,
ffiffiffiffi ffi
k 2
p χ
α
ð Þ K 2
ð Þ, Á Á Á,
ffiffiffiffiffiffi
k n c
p χ
α
ð Þ K n c
ð Þ
n
o
:
ð18:140Þ
18.6 Classes and Irreducible Representations
709