P
α
ð Þ
X d α
l¼1
P
α
ð Þ
l l
ð Þ :
ð18:173Þ
Similarly, P
(α) is again a projection operator as well.
From (18.156), P
(α) in (18.173) can be rewritten as
P
α
ð Þ
¼
d α
n
X
g
X d α
l¼1
D
α
ð Þ
ll g
ð Þ
à g ¼
d α
n
X
g
χ
α
ð Þ g
ð Þ
h
i Ã
g:
ð18:174Þ
Returning to (18.155) and taking summation over l there, we have
X d α
l¼1
P
α
ð Þ
l l
ð Þ f ¼
X d α
l¼1
c
α
ð Þ
l ψ
α
ð Þ
l :
Using (18.173), we get
P
α
ð Þ f ¼
X d α
l¼1
c
α
ð Þ
l ψ
α
ð Þ
l :
ð18:175Þ
Thus, an operator P
(α) has a clear meaning. That is, P
(α) plays a role in extracting
all the vectors (or functions) including their coefficients. Defining those functions as
ψ
α
ð Þ
X d α
l¼1
c
α
ð Þ
l ψ
α
ð Þ
l ,
ð18:176Þ
we succinctly rewrite (18.175) as
P
α
ð Þ f ¼ ψ
α
ð Þ
:
ð18:177Þ
In turn, let us calculate P
(β) P
(α) . This can be done with (18.174) as follows:
P
β
ð Þ P
α
ð Þ
¼
d β
n
X
g
χ
β
ð Þ g
ð Þ
h
i Ã
g
d α
n
X
g 0 χ
α
ð Þ g
0
ð Þ
h
i Ã
g
0
:
ð18:178Þ
To carry out the calculation, (i) first we replace g
0 with g
À1 g
0 and rewrite the
summation over g
0 as that over g
À1 g
0 . (ii) Using the homomorphism and unitarity of
the representation, we rewrite [χ
(α) (g
0 )]
à as
χ
α
ð Þ g
0
ð Þ
h
i à ¼
X
i,j
D
α
ð Þ g
ð Þ
i
ji
h
h
D
α
ð Þ g
0
ð Þ
i
Ã
ji :
ð18:179Þ
Rewriting (18.178) we have
718
18 Representation Theory of Groups