Appendix G
Spinor Representations
Contents
G.1 CharacterTables ........................................................... 235
G.2 Subduction ................................................................ 237
G.3 Canonical-Basis Relationships ............................................... 237
G.4 Direct-Product Tables ......................................................
240
G.5 Coupling Coefficients ......................................................
241
Extensive character tables for double groups were provided by Herzberg. The
ℵ symbol in the present table corresponds to the Bethe rotation through an angle
of 2π . Spin-orbit coupling coefficients for the icosahedral double group have been
listed by Fowler and Ceulemans [12]. The notation ρ 1 ,ρ 2 for conjugate components
follows Griffith. The single-valued irreps in Appendix A also represent the double
groups. The rotation through 2π leaves these irreps invariant. Their characters under
ˆ
R and ℵ ˆ
R are thus the same.
G.1 Character Tables
D ∗
2
ˆ
E
ˆ
ℵ
2 ˆ
C
z
2
2 ˆ
C
y
2
2 ˆ
C x
2
E 1/2 (Γ 5 )
2
−20
0
0
D ∗
3
ˆ
E
ˆ
ℵ
2 ˆ
C 3
2 ˆ
ℵ ˆ
C 3
3 ˆ
C 2
3 ˆ
ℵ ˆ
C 2
E 1/2 (Γ 4 )
2
−21−10
0
E 3/2
ρ 1
1
−1
−11
i
−i
ρ 2
1
−1
−11
−ii
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5,
© Springer Science+Business Media Dordrecht 2013
235