In fact, R is a representation. This is confirmed as follows: In (18.85), if
g
À1
i g ν g j ¼ e, δ g
À1
i g ν g j
À
Á ¼ 1. This occurs when g ν g j ¼ g i (A). Meanwhile, let us
consider a situation where g
À1
j g μ g k ¼ e. This occurs when g μ g k ¼ g j (B). Replacing
g j in (A) with that in (B), we have
g ν g μ g k ¼ g i :
ð18:88Þ
That is,
g
À1
i g ν g μ g k ¼ e:
ð18:89Þ
If we choose g i and g ν , g j is uniquely decided from (A). If g μ is separately chosen,
then g k is uniquely decide from (B) as well, because g j has already been uniquely
decided. Thus, performing the following matrix calculations, we get
X
j
D
R
ð Þ g ν
ð Þ
i
ij
h
h
D
R
ð Þ g μ
À Á i
jk ¼
X
j
δ g
À1
i g ν g j
δ g
À1
j g μ g k
¼ δ g
À1
i g ν g μ g k
À
Á ¼ D
R
ð Þ g ν g μ
À
Á
h
i
ik
:
ð18:90Þ
Rewriting (18.90) in a matrix product form, we get
D
R
ð Þ g ν
ð ÞD
R
ð Þ g μ
À Á ¼ D
R
ð Þ g ν g μ
À
Á :
ð18:91Þ
Thus, D
(R) is certainly a representation.
To further confirm this, let us think of an example.
Example 18.2 Let us consider a thiophene molecule that we have already examined
in Sect. 17.2. In a multiplication table, we arrange E, C 2 , σ v , σ v 0 in a first column and
their inverse element E, C 2 , σ v , σ v 0 in a first row. In this case, the inverse element is
the same as original element itself. Paying attention, e.g., to C 2 , we allocate the
number 1 on the place where C 2 appears and the number 0 otherwise. Then, that
matrix is a regular representation of C 2 ; see Table 18.3. Thus as D
(R) (C 2 ), we get
D
R
ð Þ C 2
ð Þ ¼
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
0
B
B
B
@
1
C
C
C
A
:
ð18:92Þ
As evidenced in (18.92), the rearrangement theorem ensures that the number
1 appears once and only once in each column and each row in such a way that
individual column and row vectors become linearly independent. Thus, at the same
time, we confirm that the matrix is unitary.
18.5 Regular Representation and Group Algebra
701
g
À1
i g ν g j ¼ e, δ g
À1
i g ν g j
À
Á ¼ 1. This occurs when g ν g j ¼ g i (A). Meanwhile, let us
consider a situation where g
À1
j g μ g k ¼ e. This occurs when g μ g k ¼ g j (B). Replacing
g j in (A) with that in (B), we have
g ν g μ g k ¼ g i :
ð18:88Þ
That is,
g
À1
i g ν g μ g k ¼ e:
ð18:89Þ
If we choose g i and g ν , g j is uniquely decided from (A). If g μ is separately chosen,
then g k is uniquely decide from (B) as well, because g j has already been uniquely
decided. Thus, performing the following matrix calculations, we get
X
j
D
R
ð Þ g ν
ð Þ
i
ij
h
h
D
R
ð Þ g μ
À Á i
jk ¼
X
j
δ g
À1
i g ν g j
δ g
À1
j g μ g k
¼ δ g
À1
i g ν g μ g k
À
Á ¼ D
R
ð Þ g ν g μ
À
Á
h
i
ik
:
ð18:90Þ
Rewriting (18.90) in a matrix product form, we get
D
R
ð Þ g ν
ð ÞD
R
ð Þ g μ
À Á ¼ D
R
ð Þ g ν g μ
À
Á :
ð18:91Þ
Thus, D
(R) is certainly a representation.
To further confirm this, let us think of an example.
Example 18.2 Let us consider a thiophene molecule that we have already examined
in Sect. 17.2. In a multiplication table, we arrange E, C 2 , σ v , σ v 0 in a first column and
their inverse element E, C 2 , σ v , σ v 0 in a first row. In this case, the inverse element is
the same as original element itself. Paying attention, e.g., to C 2 , we allocate the
number 1 on the place where C 2 appears and the number 0 otherwise. Then, that
matrix is a regular representation of C 2 ; see Table 18.3. Thus as D
(R) (C 2 ), we get
D
R
ð Þ C 2
ð Þ ¼
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
0
B
B
B
@
1
C
C
C
A
:
ð18:92Þ
As evidenced in (18.92), the rearrangement theorem ensures that the number
1 appears once and only once in each column and each row in such a way that
individual column and row vectors become linearly independent. Thus, at the same
time, we confirm that the matrix is unitary.
18.5 Regular Representation and Group Algebra
701