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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/19j/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 19J / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

Over

C

, the operator representing

r

has an eigenvector

v

with eigenvalue

λ \neq = 0

. The relation

t r t^{- 1} = r^{- 1}

gives

r (t v) = t (r^{- 1} v) = λ^{- 1} t v .

(1)

Therefore $span {v, t v}$ is a nonzero $G$ -invariant subspace of dimension at most two. If $V$ is irreducible it equals this subspace, so $dim V \leq 2$ .

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