with vectors mostly in ℝ
3 . Therefore, vectors naturally possess geometric features.
In this section, we extend a notion of vectors so that they can be treated under a wider
scope. More specifically, we include mathematical functions treated in analysis as
vectors.
To this end, let us think of a basis of a representation. According to the dimension
d of the representation, we have d basis vectors. Here we adopt d linearly independent basis vectors for the representation.
Let ψ 1 , ψ 2 , Á Á Á, and ψ d be linearly independent vectors in a vector space V
d . Let
ℊ ¼ {g 1 , g 2 , Á Á Á, g n } be a finite group of order n. Here we assume that g i (1 i n)
is a linear transformation (or operator) such as a symmetry operation dealt with in
Chap. 17. Suppose that the following relation holds with g i 2 ℊ (1 i n):
g i ψ ν
ð Þ ¼
X d
μ¼1
ψ μ D μν g i
ð Þ 1 ν d
ð
Þ :
ð18:22Þ
Here we followed the notation of (11.37) that represented a linear transformation
of a vector. Rewriting (18.22) more explicitly, we have
g i ψ ν
ð Þ ¼ ψ 1 ψ 2 Á Á Áψ d
ð
Þ
D 11 g i
ð Þ Á Á Á D 1d g i
ð Þ
⋮
⋱
⋮
D d1 g i
ð Þ Á Á Á D dd g i
ð Þ
0
B
@
1
C
A:
ð18:23Þ
Comparing (18.23) with (11.37), we notice that ψ 1 , ψ 2 , Á Á Á, and ψ d act as vectors.
The corresponding coordinates of the vectors (or a column vector); i.e., x 1 , x 2 , Á Á Á,
and x d have been omitted.
Now let us make sure that a set L consisting of {D(g 1 ), D(g 2 ), Á Á Á, D(g n )} forms a
representation. Operating g j on (18.22), we have
g j g i ψ ν
ð Þ ¼
½g j g i
Â
à ψ ν
ð Þ ¼ g j
X d
μ¼1
ψ μ D μν g i
ð Þ ¼
X d
μ¼1
g j ψ μ D μν g i
ð Þ
Â
Ã
¼
X d
μ¼1
g j ψ μ
À
Á
D μν g i
ð Þ ¼
X d
μ¼1
X d
λ¼1
ψ λ D λμ g j
À Á
D μν g i
ð Þ
¼
X d
λ¼1
ψ λ D g j
À Á
D g i
ð Þ
Â
Ã
λν
:
ð18:24Þ
Putting g j g i ¼ g k according to multiplication of group elements, we get
g k ψ ν
ð Þ ¼
X d
λ¼1
ψ λ D g j
À Á
D g i
ð Þ
Â
Ã
λν
:
ð18:25Þ
Meanwhile, replacing g i in (18.22) with g k , we have
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18 Representation Theory of Groups
3 . Therefore, vectors naturally possess geometric features.
In this section, we extend a notion of vectors so that they can be treated under a wider
scope. More specifically, we include mathematical functions treated in analysis as
vectors.
To this end, let us think of a basis of a representation. According to the dimension
d of the representation, we have d basis vectors. Here we adopt d linearly independent basis vectors for the representation.
Let ψ 1 , ψ 2 , Á Á Á, and ψ d be linearly independent vectors in a vector space V
d . Let
ℊ ¼ {g 1 , g 2 , Á Á Á, g n } be a finite group of order n. Here we assume that g i (1 i n)
is a linear transformation (or operator) such as a symmetry operation dealt with in
Chap. 17. Suppose that the following relation holds with g i 2 ℊ (1 i n):
g i ψ ν
ð Þ ¼
X d
μ¼1
ψ μ D μν g i
ð Þ 1 ν d
ð
Þ :
ð18:22Þ
Here we followed the notation of (11.37) that represented a linear transformation
of a vector. Rewriting (18.22) more explicitly, we have
g i ψ ν
ð Þ ¼ ψ 1 ψ 2 Á Á Áψ d
ð
Þ
D 11 g i
ð Þ Á Á Á D 1d g i
ð Þ
⋮
⋱
⋮
D d1 g i
ð Þ Á Á Á D dd g i
ð Þ
0
B
@
1
C
A:
ð18:23Þ
Comparing (18.23) with (11.37), we notice that ψ 1 , ψ 2 , Á Á Á, and ψ d act as vectors.
The corresponding coordinates of the vectors (or a column vector); i.e., x 1 , x 2 , Á Á Á,
and x d have been omitted.
Now let us make sure that a set L consisting of {D(g 1 ), D(g 2 ), Á Á Á, D(g n )} forms a
representation. Operating g j on (18.22), we have
g j g i ψ ν
ð Þ ¼
½g j g i
Â
à ψ ν
ð Þ ¼ g j
X d
μ¼1
ψ μ D μν g i
ð Þ ¼
X d
μ¼1
g j ψ μ D μν g i
ð Þ
Â
Ã
¼
X d
μ¼1
g j ψ μ
À
Á
D μν g i
ð Þ ¼
X d
μ¼1
X d
λ¼1
ψ λ D λμ g j
À Á
D μν g i
ð Þ
¼
X d
λ¼1
ψ λ D g j
À Á
D g i
ð Þ
Â
Ã
λν
:
ð18:24Þ
Putting g j g i ¼ g k according to multiplication of group elements, we get
g k ψ ν
ð Þ ¼
X d
λ¼1
ψ λ D g j
À Á
D g i
ð Þ
Â
Ã
λν
:
ð18:25Þ
Meanwhile, replacing g i in (18.22) with g k , we have
684
18 Representation Theory of Groups