g k ψ ν
ð Þ ¼
X d
μ¼1
ψ λ D λν g k
ð Þ:
ð18:26Þ
Comparing (18.25) and (18.26) and considering the uniqueness of vector representation based on linear independence of the basis vectors (see Sect. 11.1), we get
D g j
À Á
D g i
ð Þ
Â
Ã
λν
¼ D λν g k
ð Þ D g k
ð Þ
½
λν ,
ð18:27Þ
where the last identity follows the notation (11.38) of Sect. 11.2. Hence, we have
D g j
À Á
D g i
ð Þ ¼ D g k
ð Þ:
ð18:28Þ
Thus, the set L consisting of {D(g 1 ), D(g 2 ), Á Á Á, D(g n )} is certainly a representation of the group ℊ ¼ {g 1 , g 2 , Á Á Á, g n }. In such a case, the set B consisting of linearly
independent d functions (i.e., vectors)
B ¼ ψ 1 , ψ 2 , Á Á Á, ψ d
f
g
ð18:29Þ
is said to be basis functions of the representation D. The number d equals the
dimension of representation. Correspondingly, the representation matrix is a (d, d )
square matrix. As remarked above, the correspondence between the elements
of L and ℊ is not necessarily one-to-one (isomorphic), but may be n-to-one
(homomorphic).
Definition 18.2 Let D and D
0 be two representations of ℊ ¼ {g 1 , g 2 , Á Á Á, g n }.
Suppose that these representations are related to each other by similarity transformation such that
D
0 g i
ð Þ ¼ T
À1 D g i
ð ÞT 1 i n
ð
Þ ,
ð18:30Þ
where T is a non-singular matrix. Then, D and D
0 are said to be equivalent representations, or simply equivalent. If the representations are not equivalent, they are
called inequivalent.
Suppose that B ¼ ψ 1 , ψ 2 , Á Á Á, ψ d
f
g is a basis of a representation D of
ℊ ¼ {g 1 , g 2 , Á Á Á, g n }. Then we have (18.22). Let T be a non-singular matrix. Using
T, we want to transform a basis of the representation D from B to a new set
B
0
¼ ψ
0
1 , ψ
0
2 , Á Á Á, ψ
0
d
È
É
. Individual elements of the new basis are expressed as
ψ
0
ν ¼
X d
λ¼1
ψ λ T λν :
ð18:31Þ
Since T is non-singular, this ensures that ψ
0
1 , ψ
0
2 , Á Á Á, and ψ
0
d are linearly independent and, hence, that B
0 forms another basis set of D (see Discussion of Sect.
11.4). Thus, we can describe ψ μ (1 μ n) in terms of ψ
0
ν . That is
18.2 Basis Functions of Representation
685
ð Þ ¼
X d
μ¼1
ψ λ D λν g k
ð Þ:
ð18:26Þ
Comparing (18.25) and (18.26) and considering the uniqueness of vector representation based on linear independence of the basis vectors (see Sect. 11.1), we get
D g j
À Á
D g i
ð Þ
Â
Ã
λν
¼ D λν g k
ð Þ D g k
ð Þ
½
λν ,
ð18:27Þ
where the last identity follows the notation (11.38) of Sect. 11.2. Hence, we have
D g j
À Á
D g i
ð Þ ¼ D g k
ð Þ:
ð18:28Þ
Thus, the set L consisting of {D(g 1 ), D(g 2 ), Á Á Á, D(g n )} is certainly a representation of the group ℊ ¼ {g 1 , g 2 , Á Á Á, g n }. In such a case, the set B consisting of linearly
independent d functions (i.e., vectors)
B ¼ ψ 1 , ψ 2 , Á Á Á, ψ d
f
g
ð18:29Þ
is said to be basis functions of the representation D. The number d equals the
dimension of representation. Correspondingly, the representation matrix is a (d, d )
square matrix. As remarked above, the correspondence between the elements
of L and ℊ is not necessarily one-to-one (isomorphic), but may be n-to-one
(homomorphic).
Definition 18.2 Let D and D
0 be two representations of ℊ ¼ {g 1 , g 2 , Á Á Á, g n }.
Suppose that these representations are related to each other by similarity transformation such that
D
0 g i
ð Þ ¼ T
À1 D g i
ð ÞT 1 i n
ð
Þ ,
ð18:30Þ
where T is a non-singular matrix. Then, D and D
0 are said to be equivalent representations, or simply equivalent. If the representations are not equivalent, they are
called inequivalent.
Suppose that B ¼ ψ 1 , ψ 2 , Á Á Á, ψ d
f
g is a basis of a representation D of
ℊ ¼ {g 1 , g 2 , Á Á Á, g n }. Then we have (18.22). Let T be a non-singular matrix. Using
T, we want to transform a basis of the representation D from B to a new set
B
0
¼ ψ
0
1 , ψ
0
2 , Á Á Á, ψ
0
d
È
É
. Individual elements of the new basis are expressed as
ψ
0
ν ¼
X d
λ¼1
ψ λ T λν :
ð18:31Þ
Since T is non-singular, this ensures that ψ
0
1 , ψ
0
2 , Á Á Á, and ψ
0
d are linearly independent and, hence, that B
0 forms another basis set of D (see Discussion of Sect.
11.4). Thus, we can describe ψ μ (1 μ n) in terms of ψ
0
ν . That is
18.2 Basis Functions of Representation
685