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 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2022 / 1 / a / Solution

Past exam of the groups and representations course of the University of Oxford / 2022 / 1 / b / Solution 2026-10-25

In a one-dimensional representation, the scalar images of

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

, so they are inequivalent. The sum of squares of irreducible representation dimensions is already

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

, proving completeness.

Put

s = 2 cos (2 π /5) = (5 - 1) /2

and

t = 2 cos (4 π /5) = - (5 + 1) /2

. Taking traces yields the character table

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

also verify character orthogonality directly.

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