X
g
D il g
ð ÞD jm g
ð Þ
à ¼ λ lm δ ij :
ð18:58Þ
Next, we wish to determine coefficients λ lm . To this end, setting i ¼ j and
summing over i in (18.57), we get for LHS
X
g
X
i
D il g
ð ÞD mi g
À1
À Á ¼
X
g
D g
À1
À Á
D g
ð Þ
!
ml
¼
X
g
D g
À1 g
À
Á
!
ml
¼
X
g
D e
ð Þ
½
Š ml ¼
X
g
δ ml ¼ nδ ml ,
ð18:59Þ
where n is equal to the order of group. As for RHS, we have
X
i
λ lm δ ii ¼ λ lm d,
ð18:60Þ
where d is equal to a dimension of D. From (18.59) and (18.60), we get
λ lm d ¼ nδ lm or λ lm ¼
n
d
δ lm :
ð18:61Þ
Therefore, from (18.58)
X
g
D il g
ð ÞD jm g
ð Þ
à ¼
n
d
δ lm δ ij :
ð18:62Þ
Specifying a species of the irreducible representation, we get
X
g
D
α
ð Þ
il g
ð ÞD
α
ð Þ
jm g
ð Þ
à ¼
n
d α
δ lm δ ij ,
ð18:63Þ
where d α is a dimension of D
(α) .
Next, we examine the relationship between two inequivalent irreducible representations. Let D
(α) and D
(β) be such representations with dimensions d α and d β ,
respectively. Let us construct a matrix B such that
B ¼
X
g
D
α
ð Þ g
ð ÞXD
β
ð Þ g
À1
À Á
,
ð18:64Þ
where X is again an arbitrary matrix. Hence,
696
18 Representation Theory of Groups