Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 1 / b

Table of contents
- Solution b

Solution

The assumed one-dimensional-image property gives

b_{t} e (s) = c e (t)

for some

c \in F

. The coefficient of

{t}

in

e (t)

is one, so

c = ⟨ b_{t} e (s), {t}⟩ .

The tabloid bilinear form is invariant, and the involution on the group algebra fixes the Column antisymmetrizer of a Young tableau because inversion preserves sign. Therefore

c = ⟨ e (s), b_{t} {t}⟩ = ⟨ e (s), e (t)⟩,

which proves the formula.

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