P
α
ð Þ
l l
ð Þ cc
α
ð Þ
l ψ
α
ð Þ
l þ dd
α
ð Þ
l ϕ
α
ð Þ
l
h
i
¼ cP
α
ð Þ
l l
ð Þ c
α
ð Þ
l ψ
α
ð Þ
l þ dP
α
ð Þ
l l
ð Þ d
α
ð Þ
l ϕ
α
ð Þ
l
¼ cc
α
ð Þ
l ψ
α
ð Þ
l þ dd
α
ð Þ
l ϕ
α
ð Þ
l ,
ð18:165Þ
where with the last equality we used (18.164). This means that an arbitrary linear
combination of c
α
ð Þ
l ψ
α
ð Þ
l and d
α
ð Þ
l ϕ
α
ð Þ
l again belongs to the position l of an irreducible
representation α. If ψ
α
ð Þ
l
and ϕ
α
ð Þ
l
are linearly independent, we can construct two
orthonormal basis vectors following Theorem 13.2 (Gram–Schmidt
orthonormalization theorem). If there are other linearly independent vectors
p
α
ð Þ
l ξ
α
ð Þ
l , q
α
ð Þ
l φ
α
ð Þ
l , Á Á Á, where ξ
α
ð Þ
l , φ
α
ð Þ
l , etc. belong to the position l of an irreducible
representation α, then cc
α
ð Þ
l ψ
α
ð Þ
l þ dd
α
ð Þ
l ϕ
α
ð Þ
l þ pp
α
ð Þ
l ξ
α
ð Þ
l þ qq
α
ð Þ
l φ
α
ð Þ
l þ Á Á Á again
belongs to the position l of an irreducible representation α. Thus, we can construct
orthonormal basis vectors of the representation space according to Theorem 13.2.
Regarding arbitrary functions f and h as vectors and using (18.160), we make an
inner product of (18.160) such that
h
h P
α
ð Þ
l l
ð Þ P
β
ð Þ
s s
ð Þ f
j i ¼ h
h P
α
ð Þ
l l
ð Þ P
α
ð Þ
l l
ð Þ P
β
ð Þ
s s
ð Þ
f
j i ¼ h
h P
α
ð Þ
l l
ð Þ δ αβ δ ls P
α
ð Þ
l s
ð Þ
f i
¼ δ αβ δ ls hP
α
ð Þ
l l
ð Þ P
α
ð Þ
l s
ð Þ
f
D
E
,
ð18:166Þ
where with the first equality we used P
α
ð Þ
l l
ð Þ ¼ P
α
ð Þ
l l
ð Þ P
α
ð Þ
l l
ð Þ (18.154). Meanwhile, from
(18.148) we have
P
α
ð Þ
l s
ð Þ j f i ¼ c
α
ð Þ
s ψ
α
ð Þ
l :
ð18:167Þ
Also using (18.155), we get
P
α
ð Þ
l l
ð Þ j hi ¼ d
α
ð Þ
l
j ϕ
α
ð Þ
l i:
ð18:168Þ
Taking adjoint of (18.168), we have
h P
α
ð Þ
l l
ð Þ
h
i
{ ¼ hh P
α
ð Þ
l l
ð Þ
¼ d
α
ð Þ
l
h
i à hϕ
α
ð Þ
l
j ,
ð18:169Þ
where we used P
α
ð Þ
l l
ð Þ
h
i { ¼ P
α
ð Þ
l l
ð Þ (18.157). The relation (18.169) is due to the notation
of Sect. 13.3. Substituting (18.167) and (18.168) for (18.166),
h P
α
ð Þ
l l
ð Þ P
β
ð Þ
s s
ð Þ
f
D
E
¼ δ αβ δ ls d
α
ð Þ
l
h
i Ã
c
α
ð Þ
s
ϕ
α
ð Þ
l jψ
α
ð Þ
l
D
E
:
Meanwhile, (18.166) can also be described as
716
18 Representation Theory of Groups
α
ð Þ
l l
ð Þ cc
α
ð Þ
l ψ
α
ð Þ
l þ dd
α
ð Þ
l ϕ
α
ð Þ
l
h
i
¼ cP
α
ð Þ
l l
ð Þ c
α
ð Þ
l ψ
α
ð Þ
l þ dP
α
ð Þ
l l
ð Þ d
α
ð Þ
l ϕ
α
ð Þ
l
¼ cc
α
ð Þ
l ψ
α
ð Þ
l þ dd
α
ð Þ
l ϕ
α
ð Þ
l ,
ð18:165Þ
where with the last equality we used (18.164). This means that an arbitrary linear
combination of c
α
ð Þ
l ψ
α
ð Þ
l and d
α
ð Þ
l ϕ
α
ð Þ
l again belongs to the position l of an irreducible
representation α. If ψ
α
ð Þ
l
and ϕ
α
ð Þ
l
are linearly independent, we can construct two
orthonormal basis vectors following Theorem 13.2 (Gram–Schmidt
orthonormalization theorem). If there are other linearly independent vectors
p
α
ð Þ
l ξ
α
ð Þ
l , q
α
ð Þ
l φ
α
ð Þ
l , Á Á Á, where ξ
α
ð Þ
l , φ
α
ð Þ
l , etc. belong to the position l of an irreducible
representation α, then cc
α
ð Þ
l ψ
α
ð Þ
l þ dd
α
ð Þ
l ϕ
α
ð Þ
l þ pp
α
ð Þ
l ξ
α
ð Þ
l þ qq
α
ð Þ
l φ
α
ð Þ
l þ Á Á Á again
belongs to the position l of an irreducible representation α. Thus, we can construct
orthonormal basis vectors of the representation space according to Theorem 13.2.
Regarding arbitrary functions f and h as vectors and using (18.160), we make an
inner product of (18.160) such that
h
h P
α
ð Þ
l l
ð Þ P
β
ð Þ
s s
ð Þ f
j i ¼ h
h P
α
ð Þ
l l
ð Þ P
α
ð Þ
l l
ð Þ P
β
ð Þ
s s
ð Þ
f
j i ¼ h
h P
α
ð Þ
l l
ð Þ δ αβ δ ls P
α
ð Þ
l s
ð Þ
f i
¼ δ αβ δ ls hP
α
ð Þ
l l
ð Þ P
α
ð Þ
l s
ð Þ
f
D
E
,
ð18:166Þ
where with the first equality we used P
α
ð Þ
l l
ð Þ ¼ P
α
ð Þ
l l
ð Þ P
α
ð Þ
l l
ð Þ (18.154). Meanwhile, from
(18.148) we have
P
α
ð Þ
l s
ð Þ j f i ¼ c
α
ð Þ
s ψ
α
ð Þ
l :
ð18:167Þ
Also using (18.155), we get
P
α
ð Þ
l l
ð Þ j hi ¼ d
α
ð Þ
l
j ϕ
α
ð Þ
l i:
ð18:168Þ
Taking adjoint of (18.168), we have
h P
α
ð Þ
l l
ð Þ
h
i
{ ¼ hh P
α
ð Þ
l l
ð Þ
¼ d
α
ð Þ
l
h
i à hϕ
α
ð Þ
l
j ,
ð18:169Þ
where we used P
α
ð Þ
l l
ð Þ
h
i { ¼ P
α
ð Þ
l l
ð Þ (18.157). The relation (18.169) is due to the notation
of Sect. 13.3. Substituting (18.167) and (18.168) for (18.166),
h P
α
ð Þ
l l
ð Þ P
β
ð Þ
s s
ð Þ
f
D
E
¼ δ αβ δ ls d
α
ð Þ
l
h
i Ã
c
α
ð Þ
s
ϕ
α
ð Þ
l jψ
α
ð Þ
l
D
E
:
Meanwhile, (18.166) can also be described as
716
18 Representation Theory of Groups