Spinor Representations - Group Theory Applied to Chemistry - page 232

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 2 π . 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 2 π leaves these irreps invariant. Their characters under
R and

ℵ R are thus the same.

G.1 Character Tables

2 C 2

E ℵ

2 C 2 2 C 2

D 2

E 1 / 2 (Γ 5 ) 2 − 2 0 0 0

E ℵ

2 C 3 2 ℵ C 3 3 C 2 3 ℵ C 2

D 3

E 1 / 2 (Γ 4 ) 2 − 2 1 − 1 0 0

ρ 1

E 3 / 2 1 − 1 − 1 1 i − i

ρ 2 1 − 1 − 1 1 − i i