where D
(W ) (g i ) and D
W
⊥
ð
Þ g i
ð Þ are representation matrices associated with subspaces
W and W
⊥
, respectively. This is the converse of the additive representations that
appeared in Definition 18.3. From (11.17) and (11.18), we have
dimV ¼ dimW þ dimW
⊥
:
ð18:37Þ
If V is a d-dimensional vector space, V is spanned by d linearly independent basis
vectors (or functions) ψ μ (1 μ d). Suppose that the dimension W and W
⊥ is d
(W )
and d
W
⊥
ð
Þ , respectively. Then, W and W
⊥ are spanned by d
(W ) linearly independent
vectors and d
W
⊥
ð
Þ linearly independent vectors, respectively. The subspaces W and
W
⊥ may well further be decomposed into orthogonal complements [i.e., D(g i )invariant subspaces of V]. In this situation, in general, we write
D g i
ð Þ ¼ D
1
ð Þ g i
ð Þ È D
2
ð Þ g i
ð Þ È Á Á Á È D
ω
ð Þ g i
ð Þ:
ð18:38Þ
We will develop further detailed discussion in Sect. 18.4.
If the aforementioned decomposition cannot be done, D is irreducible. Therefore,
it will be of great importance to examine a dimension of irreducible representations
contained in (18.38). This is closely related to the choice of basis vectors and
properties of the invariant subspaces. Notice that the same irreducible representation
may repeatedly occur in (18.38).
Before advancing to the next section, however, let us think of an example to get
used to abstract concepts. This is at the same time for the purpose of taking the
contents of Chap. 19 in advance.
Example 18.1 Figure 18.1 shows structural formulae including resonance structures for allyl radical. Thanks to the resonance structures, the allyl radical belongs to
C 2v ; see the multiplication table in Table 17.1. The molecule lies on the yz-plane and
a line connecting C 1 and a bonded H is the C 2 axis (see Fig. 18.1). The C 2 axis is
identical to the z-axis. The molecule has mirror symmetries with respect to the yzand zx-planes. We denote π-orbitals of C 1 , C 2 , and C 3 by ϕ 1 , ϕ 2 , and ϕ 3 , respectively. We suppose that these orbitals extend toward the x-direction with a positive
sign in the upper side and a negative sign in the lower side relative to the plane of
paper. Notice that we follow custom of the group theory notation with the coordinate
setting in Fig. 18.1.
We consider an inner vector space V
3 spanned by ϕ 1 , ϕ 2 , and ϕ 3 . To explicitly
show that these are vectors of the inner vector space, we express them as jϕ 1 i, jϕ 2 i,
and jϕ 3 i in this example. Then, according to (11.19) of Sect. 11.1, we write
V
3
¼ Span jϕ 1 i, jϕ 2 i, jϕ 3 i
f
g :
The vector space V
3 is a representation space pertinent to a representation D of the
present example. Also, in parallel to (13.32) of Sect. 13.2 we express an arbitrary
vector jψi 2 V
3 as
688
18 Representation Theory of Groups
(W ) (g i ) and D
W
⊥
ð
Þ g i
ð Þ are representation matrices associated with subspaces
W and W
⊥
, respectively. This is the converse of the additive representations that
appeared in Definition 18.3. From (11.17) and (11.18), we have
dimV ¼ dimW þ dimW
⊥
:
ð18:37Þ
If V is a d-dimensional vector space, V is spanned by d linearly independent basis
vectors (or functions) ψ μ (1 μ d). Suppose that the dimension W and W
⊥ is d
(W )
and d
W
⊥
ð
Þ , respectively. Then, W and W
⊥ are spanned by d
(W ) linearly independent
vectors and d
W
⊥
ð
Þ linearly independent vectors, respectively. The subspaces W and
W
⊥ may well further be decomposed into orthogonal complements [i.e., D(g i )invariant subspaces of V]. In this situation, in general, we write
D g i
ð Þ ¼ D
1
ð Þ g i
ð Þ È D
2
ð Þ g i
ð Þ È Á Á Á È D
ω
ð Þ g i
ð Þ:
ð18:38Þ
We will develop further detailed discussion in Sect. 18.4.
If the aforementioned decomposition cannot be done, D is irreducible. Therefore,
it will be of great importance to examine a dimension of irreducible representations
contained in (18.38). This is closely related to the choice of basis vectors and
properties of the invariant subspaces. Notice that the same irreducible representation
may repeatedly occur in (18.38).
Before advancing to the next section, however, let us think of an example to get
used to abstract concepts. This is at the same time for the purpose of taking the
contents of Chap. 19 in advance.
Example 18.1 Figure 18.1 shows structural formulae including resonance structures for allyl radical. Thanks to the resonance structures, the allyl radical belongs to
C 2v ; see the multiplication table in Table 17.1. The molecule lies on the yz-plane and
a line connecting C 1 and a bonded H is the C 2 axis (see Fig. 18.1). The C 2 axis is
identical to the z-axis. The molecule has mirror symmetries with respect to the yzand zx-planes. We denote π-orbitals of C 1 , C 2 , and C 3 by ϕ 1 , ϕ 2 , and ϕ 3 , respectively. We suppose that these orbitals extend toward the x-direction with a positive
sign in the upper side and a negative sign in the lower side relative to the plane of
paper. Notice that we follow custom of the group theory notation with the coordinate
setting in Fig. 18.1.
We consider an inner vector space V
3 spanned by ϕ 1 , ϕ 2 , and ϕ 3 . To explicitly
show that these are vectors of the inner vector space, we express them as jϕ 1 i, jϕ 2 i,
and jϕ 3 i in this example. Then, according to (11.19) of Sect. 11.1, we write
V
3
¼ Span jϕ 1 i, jϕ 2 i, jϕ 3 i
f
g :
The vector space V
3 is a representation space pertinent to a representation D of the
present example. Also, in parallel to (13.32) of Sect. 13.2 we express an arbitrary
vector jψi 2 V
3 as
688
18 Representation Theory of Groups