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

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 160 1 b
2026-09-28  0 By others on same topic  0 Discussions Create my own version
The assumed one-dimensional-image property gives
bt​e(s)=ce(t)
(1)
for some c∈F. The coefficient of {t} in e(t) is one, so
c=⟨bt​e(s),{t}⟩.
(2)
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),bt​{t}⟩=⟨e(s),e(t)⟩,
(3)
which proves the formula.

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