A reducible representation can be decomposed (or reduced) to a direct sum
of plural irreducible representations. Such an operation (or procedure) is called
reduction.
Definition 18.5 Let ℊ ¼ {g 1 , g 2 , Á Á Á, g n } be a group and let V be a vector space.
Then, we denote a representation D of ℊ operating on V by
D : ℊ ! GL V
ð Þ:
We call V a representation space (or carrier space) L S of D [3, 4].
From Definition 18.5, the dimension of representation is identical with the
dimension of V. Suppose that there is a subspace W in V. If W is D(g i )-invariant
(see Sect. 12.2) with all
8 g i 2 ℊ, W is said to be an invariant subspace of V. Here, we
say that W is D(g i )-invariant if we have | xi 2 W ⟹ D(g i )| xi 2 W; see Sect. 12.2.
Notice that | xi may well represent a function ψ ν of (18.22). In this context, we have a
following important theorem.
Theorem 18.2 [3] Let D: ℊ ! GL(V ) be a unitary representation over V. Let W be a
D(g i )-invariant subspace of V, where g i (1 i n) 2 ℊ ¼ {g 1 , g 2 , Á Á Á, g n }. Then, W
⊥
is a D(g i )-invariant subspace of V as well.
Proof Suppose that | ai 2 W
⊥ and let | bi 2 W. Then, from (13.86) we have
bjD g i
ð Þja
h
i¼ ajD g i
ð Þ
{ jb
D
E Ã ¼ aj D g i
ð Þ
½
Š
À1 jb
D
E Ã ¼ ajD g i
À1
À
Á jb
à ,
where we used the fact that D(g i ) is a unitary matrix; see (18.6). But, since W is
D(g i )-invariant from the supposition, jD(g i
À1 )| bi 2 W. Here notice that g i
À1 must be
∃ g i (1 i n) 2 ℊ. Therefore, we should have
bjD g i
ð Þja
h
i¼ ajD g i
À1
À
Á jb
à ¼ 0:
This implies that jD(g i )| ai 2 W
⊥
. This means that W
⊥ is a D(g i )-invariant
subspace of V. This completes the proof.
∎
From Theorem 14.1, we have
V ¼ W È W
⊥
:
Correspondingly, as in the case of (12.102), D(g i ) is reduced as follows:
D g i
ð Þ ¼
D
W
ð Þ g i
ð Þ
D
W
⊥
ð
Þ g i
ð Þ
!
,
18.2 Basis Functions of Representation
687