4.2 Character Theorems
57
a given group can sustain and how to analyze the irreducible contents of a given
function space. Characters are nothing other than the traces (that is, the sum of the
diagonal elements) of representation matrices. They will be represented as χ(R):
χ(R) =
i
D ii (R)
(4.23)
If the functional basis of a representation is transformed by a unitary transformation,
the trace does not change, as can easily be demonstrated. Define |f ′ =|fU. Then
the corresponding representation matrices, D ′ (R), also undergo a unitary transformation:
ˆ
R|f
′ = ˆ
R|fU =|fD(R)U
=|f
′ U
−1 D(R)U
=|f
′ D
′ (R)
(4.24)
from which it follows that
D
′ (R) = U
−1 D(R)U
(4.25)
or, for unitary U, that
D
′
ij (R) =
kl
¯
U
T
ik D kl (R)U lj
=
kl
¯
U ki D kl (R)U lj
(4.26)
The invariance of the character then follows from the orthogonality of the rows of
the unitary matrix:
χ
′ (R) =
i
D
′
ii (R)
=
kl
D kl
i
¯
U ki U li
=
kl
D kl (R)δ kl
=
k
D kk (R)
= χ(R)
(4.27)
Hence, sets of characters literally characterize representations since they are immune to the effects of unitary transformations, such as occur in Eq. (4.21) between
the complex functions |ψ +1 , |ψ −1 and the real functions |ψ x , |ψ y . The characters for the irreps are brought together in a character table. Here, the conjugacy class