REPRESENTATION THEORY
the corresponding coordinate and 2 cosθfor the two orthogonal coordinates.
If the rotation is followed by an inversion then these entries are multiplied
by−1. Atoms not transforming into themselves give a zero diagonal contri-
bution. Show that the characters of the natural representation are 12, 0, 0,
0, 2 and hence that its expression in terms of irreps is
A 1 ⊕E⊕T 1 ⊕2T 2.(b) The irreps of the bodily translational and rotational motions are included in
this expression and need to be identified and removed. Show that when this
is done it can be concluded that there are three different internal vibration
frequencies in the CH 4 molecule. State their degeneracies and check that
they are consistent with the expectednumber of normal coordinates needed
to describe the internal motions of the molecule.29.10 Investigate the properties of an alternating group and construct its character
table as follows.
(a) The set of even permutations of four objects (a proper subgroup ofS 4 )
is known as thealternating groupA 4. List its twelve members using cycle
notation.
(b) Assume that all permutations with the same cycle structure belong to the
same conjugacy class. Show that this leads to a contradiction, and hence
demonstrates that, even if two permutations have the same cycle structure,
they do not necessarily belong to the same class.
(c) By evaluating the products
p 1 = (123)(4)•(12)(34)•(132)(4) and p 2 = (132)(4)•(12)(34)•(123)(4)
deduce that the three elements ofA 4 with structure of the form (12)(34)
belong to the same class.
(d) By evaluating products of the form (1α)(βγ)•(123)(4)•(1α)(βγ), whereα, β, γ
are various combinations of 2, 3, 4, show that the class to which (123)(4)
belongs contains at least four members. Show the same for (124)(3).
(e) By combining results (b), (c) and (d) deduce thatA 4 has exactly four classes,
and determine the dimensions of its irreps.
(f) Using the orthogonality properties of characters and noting that elements of
the form (124)(3) have order 3, find the character table forA 4.29.11 Use the results of exercise 28.23 to find the character table for the dihedral group
D 5 , the symmetry group of a regular pentagon.
29.12 Demonstrate that equation (29.24) does, indeed, generate a set of vectors trans-
forming according to an irrepλ, by sketching and superposing drawings of an
equilateral triangle of springs and masses, based on that shown in figure 29.5.
(a) (b) (c)A BBA B A
CCC
30 ◦
30 ◦
Figure 29.7 The three normal vibration modes of the equilateral array. Mode
(a) is known as the ‘breathing mode’. Modes (b) and (c) transform according
to irrep E and have equal vibrational frequencies.