g ψ i ϕ j
À
Á ¼ g ψ i
ð Þg ϕ j
À Á ¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ:
ð18:203Þ
Regarding a product function ψ j ϕ i we can get a similar equation such that
g ψ j ϕ i
À
Á ¼ g ψ j
À Á
g ϕ i
ð Þ ¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
kj g
ð ÞD
α
ð Þ
li g
ð Þ:
ð18:204Þ
On the basis of the linearity of the relations (18.203) and (18.204), let us construct
a linear combination of the product functions. That is,
g ψ i ϕ j Æ ψ j ϕ i
À
Á ¼ g ψ i ϕ j
À
Á Æ g ψ j ϕ i
À
Á
¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ Æ
X
k
X
l
ψ k ϕ l D
α
ð Þ
kj g
ð ÞD
α
ð Þ
li g
ð Þ
¼
X
k
X
l
ψ k ϕ l Æ ψ l ϕ k
ð
Þ D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ:
ð18:205Þ
Here, defining Ψ
Æ
ij as
Ψ
Æ
ij ¼ ψ i ϕ j Æ ψ j ϕ i ,
ð18:206Þ
we rewrite (18.205) as
gΨ
Æ
ij ¼
X
k
X
l
Ψ
Æ
kl
1
2
D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ Æ D
α
ð Þ
li g
ð ÞD
α
ð Þ
kj g
ð Þ
h
i
n
o
:
ð18:207Þ
Notice that Ψ
þ
ij and Ψ
À
ij in (18.206) are symmetric and antisymmetric with respect
to the interchange of subscript i and j, respectively. That is,
Ψ
Æ
ij ¼ ÆΨ
Æ
ji :
ð18:208Þ
We may naturally ask how we can constitute representation (matrices) with
(18.207). To answer this question, we have to carry out calculations by replacing
g with gg
0 in (18.207) and following the procedures of Sect. 18.2. Thus, we have
gg
0
Ψ
Æ
ij ¼
X
k
X
l
Ψ
Æ
kl
1
2
D
α
ð Þ
ki gg
0
ð ÞD
α
ð Þ
lj gg
0
ð ÞÆD
α
ð Þ
li gg
0
ð ÞD
α
ð Þ
kj gg
0
ð Þ
h
i
n
o
¼
X
k
X
l
Ψ
Æ
kl
(
1
2
X
μ
D
α
ð Þ
kμ g
ð ÞD
α
ð Þ
μi g
0
ð Þ
X
ν
D
α
ð Þ
lν g
ð ÞD
α
ð Þ
νj g
0
ð Þ
h
Æ
X
μ
D
α
ð Þ
lμ g
ð ÞD
α
ð Þ
μi g
0
ð Þ
X
ν
D
α
ð Þ
kν g
ð ÞD
α
ð Þ
νj g
0
ð Þ
#)
724
18 Representation Theory of Groups
À
Á ¼ g ψ i
ð Þg ϕ j
À Á ¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ:
ð18:203Þ
Regarding a product function ψ j ϕ i we can get a similar equation such that
g ψ j ϕ i
À
Á ¼ g ψ j
À Á
g ϕ i
ð Þ ¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
kj g
ð ÞD
α
ð Þ
li g
ð Þ:
ð18:204Þ
On the basis of the linearity of the relations (18.203) and (18.204), let us construct
a linear combination of the product functions. That is,
g ψ i ϕ j Æ ψ j ϕ i
À
Á ¼ g ψ i ϕ j
À
Á Æ g ψ j ϕ i
À
Á
¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ Æ
X
k
X
l
ψ k ϕ l D
α
ð Þ
kj g
ð ÞD
α
ð Þ
li g
ð Þ
¼
X
k
X
l
ψ k ϕ l Æ ψ l ϕ k
ð
Þ D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ:
ð18:205Þ
Here, defining Ψ
Æ
ij as
Ψ
Æ
ij ¼ ψ i ϕ j Æ ψ j ϕ i ,
ð18:206Þ
we rewrite (18.205) as
gΨ
Æ
ij ¼
X
k
X
l
Ψ
Æ
kl
1
2
D
α
ð Þ
ki g
ð ÞD
α
ð Þ
lj g
ð Þ Æ D
α
ð Þ
li g
ð ÞD
α
ð Þ
kj g
ð Þ
h
i
n
o
:
ð18:207Þ
Notice that Ψ
þ
ij and Ψ
À
ij in (18.206) are symmetric and antisymmetric with respect
to the interchange of subscript i and j, respectively. That is,
Ψ
Æ
ij ¼ ÆΨ
Æ
ji :
ð18:208Þ
We may naturally ask how we can constitute representation (matrices) with
(18.207). To answer this question, we have to carry out calculations by replacing
g with gg
0 in (18.207) and following the procedures of Sect. 18.2. Thus, we have
gg
0
Ψ
Æ
ij ¼
X
k
X
l
Ψ
Æ
kl
1
2
D
α
ð Þ
ki gg
0
ð ÞD
α
ð Þ
lj gg
0
ð ÞÆD
α
ð Þ
li gg
0
ð ÞD
α
ð Þ
kj gg
0
ð Þ
h
i
n
o
¼
X
k
X
l
Ψ
Æ
kl
(
1
2
X
μ
D
α
ð Þ
kμ g
ð ÞD
α
ð Þ
μi g
0
ð Þ
X
ν
D
α
ð Þ
lν g
ð ÞD
α
ð Þ
νj g
0
ð Þ
h
Æ
X
μ
D
α
ð Þ
lμ g
ð ÞD
α
ð Þ
μi g
0
ð Þ
X
ν
D
α
ð Þ
kν g
ð ÞD
α
ð Þ
νj g
0
ð Þ
#)
724
18 Representation Theory of Groups