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 Incoming links: Conjugate Specht character

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 2 / b / iii / Solution Created 2026-09-28 Updated 2026-10-03

Conjugating a tableau exchanges row symmetrization with column antisymmetrization. The resulting module is the original Specht module twisted by the sign representation, giving the conjugate Specht character

S^{λ^{'}} ≅ S^{λ} \otimes sgn_{S_{n}} .

Taking traces yields

χ^{λ^{'}} = χ^{λ} sgn_{S_{n}} .

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