Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 1 / c / ii

Table of contents
- Solution ii

Solution

Because

C (t^{'}) = R (t)

,

e (t^{'}) = \sum_{r \in R (t)} sgn (r) {r t^{'}} .

Applying

θ

and using

re (u) = sgn (r) e (u)

gives

θ (e (t^{'})) = (\sum_{r \in R (t)} re (t)) \otimes e (u) .

Thus

m = \sum_{r \in R (t)} re (t)

. Every

r

fixes the tabloid

{t}

, while

e (t)

has coefficient one at

{t}

. Invariance of the tabloid bilinear form now yields

⟨ m, {t}⟩ = \sum_{r \in R (t)} ⟨ re (t), {t}⟩ = \sum_{r \in R (t)} 1 = ∣ R (t) ∣ .

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