178
7 Spherical Symmetry and Spins
Table 7.5 Representation
matrices for the spinor basis
in D ∗
3 . ω = exp
iπ
6
D(E) =
10
01
D(C A
2 ) =
0 −i
−i 0
D(C 3 ) =
¯
ω 2 0
0 ω 2
D(C B
2 ) =
0
¯
ω
−ω 0
D(C 2
3 ) =
−ω 2
0
0
−¯ ω 2
D(C C
2 ) =
0 −ω
¯
ω 0
Table 7.6 Character table for the double group D ∗
3
ˆ
E
ℵ
ˆ
C 3
ℵ ˆ
C 2
3
ℵ ˆ
C 3
ˆ
C 2
3
⎛
⎜
⎝
ˆ
C A
2
ℵ ˆ
C B
2
ℵ ˆ
C C
2
⎞
⎟
⎠
⎛
⎜
⎝
ℵ ˆ
C A
2
ˆ
C B
2
ˆ
C C
2
⎞
⎟
⎠
E 1/2
2
−21
−10
0
E 3/2
ρ 1
1
−1
−11i
−i
ρ 2
1
−1
−11
−ii
system is thus as follows:
ˆ
C A
2
(180
◦ , 1,
0,
0)
ˆ
C B
2
180
◦ ,
1
2
, −
√
3
2
, 0
ˆ
C C
2
180
◦ ,
1
2
,
√
3
2
, 0
(7.43)
The corresponding Cayley–Klein parameters are then determined as
ˆ
C A
2
(a = 0,b=−i)
ˆ
C B
2
a = 0,b=
√
3
2
−
i
2
ˆ
C C
2
a = 0,b=−
√
3
2
−
i
2
(7.44)
In Tables 7.5 and 7.6 we provide the corresponding matrices and the character
table. The ρ designations in the latter table refer to Kramers doublets and will be
explained in the subsequent section. The class structure of the double group in connection to the point group is determined by the Opechowski theorem [9].
Theorem 17 If a set of proper rotations, ˆ
C n , forms a class in the single group G,
then it gives rise to two separate classes in the double group, corresponding to
conjugacy classes { ˆ
C n } and {ℵ ˆ
C n }. The case of n = 2 is exceptional: when n = 2