 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2022/1/b/solution

Q

and

P

commute. The relation

PQP = Q^{- 1}

gives

R (Q)^{2} = 1

, while

R (Q)^{5} = 1

, so

R (Q) = 1

and

R (P) = \pm 1

. These give exactly

R_{0}

and

R_{1}

For

k = 1, 2

, define the two-dimensional representations

D_{k} (Q) = diag (α^{k}, α^{- k}), D_{k} (P) = (0110) .

The same invariant subspace argument as in solution proves that each is irreducible. Set

R_{2} = D_{1}

and

R_{3} = D_{2}

. Their characters differ at

Q

1^{2} + 1^{2} + 2^{2} + 2^{2} = 10

Put

s = 2 cos (2 π /5) = (5 - 1) /2

and

t = 2 cos (4 π /5) = - (5 + 1) /2

class size R_{0} R_{1} R_{2} R_{3} I 11122 Q, Q^{4} 211 s t Q^{2}, Q^{3} 211 t s Q^{k} P 51 - 1 00

The identities

s + t = - 1

s t = - 1

, and

s^{2} + t^{2} = 3

 New to topics? Read the docs here!