Mathematical Methods for Physics and Engineering - Matematica.NET

29.12 EXERCISESas the sum of two one-dimensional irreps and, using the reasoning given in the previousexample, are therefore split in frequency by the perturbation. For other values of n therepresentation is irreducible and so the degeneracy cannot be split. ◭29.12 Exercises29.1 A group G hasfourelementsI,X,Y and Z, which satisfy X 2 = Y 2 = Z 2 =XY Z = I. Show that G is Abelian and hence deduce the form of its charactertable.Show that the matrices(1 0D(I) =0 1D(Y )=(−1 −p0 1)(−1 0, D(X) =0 −1), D(Z) =(1 p0 −1where p is a real number, form a representation D of G. Find its characters anddecompose it into irreps.29.2 Using a square whose corners lie at coordinates (±1, ±1), form a natural representationof the dihedral group D 4 . Find the characters of the representation,and, using the information (and class order) in table 29.4 (p. 1102), express therepresentation in terms of irreps.Now form a representation in terms of eight 2 × 2 orthogonal matrices, byconsidering the effect of each of the elements of D 4 on a general vector (x, y).Confirm that this representation is one of the irreps found using the naturalrepresentation.29.3 The quaternion group Q (see exercise 28.20) has eight elements {±1, ±i, ±j,±k}obeying the relationsi 2 = j 2 = k 2 = −1, ij = k = −ji.Determine the conjugacy classes of Q and deduce the dimensions of its irreps.Show that Q is homomorphic to the four-element group V, which is generated bytwo distinct elements a and b with a 2 = b 2 =(ab) 2 = I. Find the one-dimensionalirreps of V and use these to help determine the full character table for Q.29.4 Construct the character table for the irreps of the permutation group S 4 asfollows.(a) By considering the possible forms of its cycle notation, determine the numberof elements in each conjugacy class of the permutation group S 4 ,andshowthat S 4 has five irreps. Give the logical reasoning that shows they must consistof two three-dimensional, one two-dimensional, and two one-dimensionalirreps.(b) By considering the odd and even permutations in the group S 4 , establish thecharacters for one of the one-dimensional irreps.(c) Form a natural matrix representation of 4 × 4 matrices based on a set ofobjects {a, b, c, d}, which may or may not be equal to each other, and, byselecting one example from each conjugacy class, show that this naturalrepresentation has characters 4, 2, 1, 0, 0. In the four-dimensional vectorspace in which each of the four coordinates takes on one of the four valuesa, b, c or d, the one-dimensional subspace consisting of the four points withcoordinates of the form {a, a, a, a} is invariant under the permutation groupand hence transforms according to the invariant irrep A 1 . The remainingthree-dimensional subspace is irreducible; use this and the characters deducedabove to establish the characters for one of the three-dimensional irreps, T 1 .1113),),