Proof Rewriting (18.108), we have
gK i ¼ K i g
8 g 2 ℊ
À
Á :
ð18:127Þ
Since a homomorphic correspondence holds between a group element and its
representation matrix, a similar correspondence holds as well with (18.127). Let b
K i
be a sum of k i matrices of the α-th irreducible representation D
(α) and let b
K i be
expressed as
b
K i ¼
X
g2K i
D
α
ð Þ
:
ð18:128Þ
Note that in (18.128) D
(α) functions as a linear transformation with respect to a
group algebra. From (18.127), we have
D
α
ð Þ b
K i ¼ b
K i D
α
ð Þ
8 g 2 ℊ
À
Á :
ð18:129Þ
Since D
(α) is an irreducible representation, b
K i must be expressed on the basis of
Schur’s Second Lemma as
b
K i ¼ λE:
ð18:130Þ
To determine λ, we take a trace of both sides of (18.130). Then, from (18.128) and
(18.130) we get
k i χ
α
ð Þ K i
ð Þ ¼ λd α :
ð18:131Þ
Thus, we get
b
K i ¼
k i
d α
χ
α
ð Þ K i
ð ÞE:
ð18:132Þ
Next, corresponding to (18.122), we have
b
K i b
K j ¼
X
l
c ijl b
K l :
ð18:133Þ
Replacing b
K l in (18.133) with that of (18.132), we get
708
18 Representation Theory of Groups