The notation using double scripts is somewhat complicated. We notice, however,
that in (18.188) the order of subscript of kilj is converted to kl, ij; i.e., the subscripts
i and l have been interchanged.
We write D
(α) and D
(β) in explicit forms as follows:
D
α
ð Þ g
ð Þ ¼
d
α
ð Þ
1,1
Á Á Á d
α
ð Þ
1,d α
⋮ ⋱
⋮
d
α
ð Þ
d α ,1
Á Á Á d
α
ð Þ
d α ,d α
0
B
@
1
C
A, D
β
ð Þ g
ð Þ ¼
d
β
ð Þ
1,1
Á Á Á d
β
ð Þ
1,d β
⋮ ⋱
⋮
d
β
ð Þ
d β ,1
Á Á Á d
β
ð Þ
d β ,d β
0
B
B
@
1
C
C
A : ð18:190Þ
Thus,
D
αÂβ
ð
Þ g
ð Þ ¼ D
α
ð Þ g
ð Þ D
β
ð Þ g
ð Þ
¼
d
α
ð Þ
1,1 D
β
ð Þ g
ð Þ Á Á Á d
α
ð Þ
1,d α
D
β
ð Þ g
ð Þ
⋮
⋱
⋮
d
α
ð Þ
d α ,1 D
β
ð Þ g
ð Þ Á Á Á d
α
ð Þ
d α ,d α
D
β
ð Þ g
ð Þ
0
B
@
1
C
A:
ð18:191Þ
To get familiar with the double-scripted notation, we describe a case of (2, 2)
matrices. Denoting
D
α
ð Þ g
ð Þ ¼
a 11 a 12
a 21 a 22
and D
β
ð Þ g
ð Þ ¼
b 11 b 12
b 21 b 22
,
ð18:192Þ
we get
D
αÂβ
ð
Þ g
ð Þ ¼ D
α
ð Þ g
ð Þ D
β
ð Þ g
ð Þ
¼
a 11 b 11
a 11 b 12 a 12 b 11 a 12 b 12
a 11 b 21
a 21 b 11
a 21 b 21
a 11 b 22 a 12 b 21 a 12 b 22
a 21 b 12 a 22 b 11 a 22 b 12
a 21 b 22 a 22 b 21 a 22 b 22
0
B
B
B
B
B
@
1
C
C
C
C
C
A
:
ð18:193Þ
Corresponding to (18.22), (18.189) describes the transformation of ψ i ϕ j regarding the double script. At the same time, a set {ψ i ϕ j ; (1 i d α , 1 j d β )} is a
basis of D
(α Â β) . In fact,
18.8 Direct-Product Representation
721