REPRESENTATION THEORY
(d) Complete the character table using orthogonality properties, and check the
summation rule for each irrep. You should obtain table 29.8.Typical element and class size
Irrep (1) (12) (123) (1234) (12)(34)
16 8 6 3
A 1 11 1 1 1
A 2 1 − 11 − 11
E 20 −10 2
T 1 31 0− 1 − 1
T 2 3 −10 1 − 1Table 29.8 The character table for the permutation groupS 4.29.5 In exercise 28.10, the group of pure rotations taking a cube into itself was found
to have 24 elements. The group is isomorphic to the permutation groupS 4 ,
considered in the previous question, and hence has the same character table, once
corresponding classes have been established. By counting the number of elements
in each class, make the correspondences below (the final two cannot be decided
purely by counting, and should be taken as given).
Permutation Symbol Action
class type (physics)
(1) I none
(123) 3 rotations about a body diagonal(12)(34) (^2) z rotation ofπabout the normal to a face
(1234) (^4) z rotations of±π/2 about the normal to a face
(12) (^2) d rotation ofπabout an axis through the
centres of opposite edges
Reformulate the character table 29.8 interms of the elements of the rotation
symmetry group (432 orO) of a cube and use it when answering exercises 29.7
and 29.8.
29.6 Consider a regular hexagon orientated so that two of its vertices lie on thex-axis.
Find matrix representations of a rotationRthrough 2π/6 and a reflectionmyin
they-axis by determining their effects on vectors lying in thexy-plane. Show
that a reflectionmxin thex-axis can be written asmx=myR^3 , and that the 12
elements of the symmetry group of the hexagon are given byRnorRnmy.
Using the representations ofRandmyas generators, find a two-dimensional
representation of the symmetry group,C 6 , of the regular hexagon. Is it a faithful
representation?
29.7 In a certain crystalline compound, a thorium atom lies at the centre of a regular
octahedron of six sulphur atoms at positions (±a, 0 ,0), (0,±a,0), (0, 0 ,±a). These
can be considered as being positioned at the centres of the faces of a cube of
side 2a. The sulphur atoms produce at the site of the thorium atom an electric
field that has the same symmetry group as a cube (432 orO).
The five degenerated-electron orbitals of the thorium atom can be expressed,
relative to any arbitrary polar axis, as
(3 cos^2 θ−1)f(r),e±iφsinθcosθf(r),e±^2 iφsin^2 θf(r).
A rotation about that polar axis by an angleφ′effectively changesφtoφ−φ′.