29.12 EXERCISES
as the sum of two one-dimensional irreps and, using the reasoning given in the previous
example, are therefore split in frequency by the perturbation. For other values ofnthe
representation is irreducible and so the degeneracy cannot be split.
29.12 Exercises29.1 A groupGhasfourelementsI,X,YandZ, which satisfyX^2 =Y^2 =Z^2 =
XY Z=I. Show thatGis Abelian and hence deduce the form of its character
table.
Show that the matrices
D(I)=(
10
01
)
, D(X)=
(
− 10
0 − 1
)
,
D(Y)=
(
− 1 −p
01)
, D(Z)=
(
1 p
0 − 1)
,
wherepis a real number, form a representationDofG. Find its characters and
decompose it into irreps.
29.2 Using a square whose corners lie at coordinates (± 1 ,±1), form a natural rep-
resentation of the dihedral groupD 4. Find the characters of the representation,
and, using the information (and class order) in table 29.4 (p. 1102), express the
representation in terms of irreps.
Now form a representation in terms of eight 2×2 orthogonal matrices, by
considering the effect of each of the elements ofD 4 on a general vector (x, y).
Confirm that this representation is one of the irreps found using the natural
representation.
29.3 The quaternion groupQ(see exercise 28.20) has eight elements{± 1 ,±i,±j,±k}
obeying the relations
i^2 =j^2 =k^2 =− 1 ,ij=k=−ji.
Determine the conjugacy classes ofQand deduce the dimensions of its irreps.
Show thatQis homomorphic to the four-element groupV, which is generated by
two distinct elementsaandbwitha^2 =b^2 =(ab)^2 =I. Find the one-dimensional
irreps ofVand use these to help determine the full character table forQ.
29.4 Construct the character table for the irreps of the permutation groupS 4 as
follows.
(a) By considering the possible forms of its cycle notation, determine the number
of elements in each conjugacy class of the permutation groupS 4 ,andshow
thatS 4 has five irreps. Give the logical reasoning that shows they must consist
of two three-dimensional, one two-dimensional, and two one-dimensional
irreps.
(b) By considering the odd and even permutations in the groupS 4 , establish the
characters for one of the one-dimensional irreps.
(c) Form a natural matrix representation of 4×4 matrices based on a set of
objects{a, b, c, d}, which may or may not be equal to each other, and, by
selecting one example from each conjugacy class, show that this natural
representation has characters 4, 2, 1, 0, 0. In the four-dimensional vector
space in which each of the four coordinates takes on one of the four values
a,b,cord, the one-dimensional subspace consisting of the four points with
coordinates of the form{a, a, a, a}is invariant under the permutation group
and hence transforms according to the invariant irrep A 1. The remaining
three-dimensional subspace is irreducible; use this and the characters deduced
above to establish the characters for one ofthe three-dimensional irreps, T 1.