= Signed Young permutation module
{title2=$\widetilde M^\lambda$}
Over the <complex numbers>, the signed Young permutation module is
$$
\widetilde M^\lambda=\operatorname{Ind}_{S_\lambda}^{S_n}\operatorname{sgn}
\cong\operatorname{sgn}\otimes M^\lambda.
$$
The isomorphism is the <tensor identity for induced representations>. The partition index is unchanged by this twist; conjugation instead changes the labels of its irreducible constituents. For the <conjugate partition> $\lambda'$, the <character inner product> of $M^\lambda$ and $\widetilde M^{\lambda'}$ is one: <Mackey restriction formula> reduces it to the unique zero-one matrix with row margins $\lambda$ and column margins $\lambda'$.
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