gΨ
þ
ij ¼
X
k
X
l
Ψ
þ
kl D
αÂα
½
Š g
ð Þ
n
o
kl,ij
:
ð18:213Þ
Then we have
gg
0
Ψ
þ
ij ¼
X
k
X
l
Ψ
þ
kl D
αÂα
½
Š gg
0
ð Þ
n
o
kl,ij
:
ð18:214Þ
Comparing (18.212) and (18.214), we finally get
D
αÂα
½
Š gg
0
ð Þ ¼ D
αÂα
½
Š g
ð ÞD
αÂα
½
Š g
0
ð Þ:
ð18:215Þ
Similarly, for the antisymmetric case, we have
gg
0
Ψ
À
ij ¼
X
k
X
l
Ψ
À
kl D
αÂα
f
g g
ð ÞD
αÂα
f
g g
0
ð Þ
È
É
kl,ij
,
ð18:216Þ
gΨ
À
ij ¼
X
k
X
l
Ψ
À
kl D
αÂα
f
g g
ð Þ
È
É
kl,ij
,
ð18:217Þ
D
αÂα
f
g gg
0
ð Þ ¼ D
αÂα
f
g g
ð ÞD
αÂα
f
g g
0
ð Þ:
ð18:218Þ
Thus, both D
[α Â α]
(g) and D
{α Â α}
(g) produce well-defined representations.
Letting dimension of the representation α be d α , we have d α (d α + 1)/2 functions
belonging to the symmetric representation and d α (d α À 1)/2 functions belonging to
the antisymmetric representation. With the two-dimensional representation, for
instance, functions belonging to the symmetric representation are
ψ 1 ϕ 1 , ψ 1 ϕ 2 þ ψ 2 ϕ 1 , and ψ 2 ϕ 2 :
A function belonging to the antisymmetric representation is
ψ 1 ϕ 2 À ψ 2 ϕ 1 :
Note that these vectors have not yet been normalized.
From (18.210) and (18.211), we can readily get useful expressions with characters of symmetric and antisymmetric representations. In (18.210) putting μ ¼ k and
ν ¼ l and summing over k and l,
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18 Representation Theory of Groups