Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 4 / c

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- Solution c

Solution

Under the subgroup in part (b),

(v, s) \in R^{4} \oplus R

(R v, s)

, hence

5 ↓ SO (4) = 4 \oplus 1 .

An element of

so_{5}

has the unique block form

X = (A - v^{T} v 0), A \in so_{4}, q u a d v \in R^{4} .

Conjugation by

ι (R)

sends

(A, v)

to

(R A R^{- 1}, R v)

. Thus

10 ↓ SO (4) = 6_{ad} \oplus 4,

and the dimensions check as

5 = 4 + 1

and

10 = 6 + 4

. These are the SO5 to SO4 branching rules.

Solved by gpt-5.6-sol high.

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