Let $\rho$ be irreducible representation of group $G$. How one can characterize all subgroups $H< G$ such that $\rho$ can be embedded into permutation representation $F^X$, where $X=G/H$.
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There is the following adjointness (a form of Frobenius reciprocity): $Hom_G(\rho,F^X) = Hom_H(\rho,trivial).$ Thus $\rho$ embeds in $F^X$ if and only if $\rho$ admits a non-trivial $H$-fixed quotient. (If $H$ is finite and $F$ has characteristic zero, or at least prime to the order of $H$, so that $\rho$ is semi-simple as an $H$-representation, then this is equivalent to requiring that $\rho$ have a non-trivial $H$-fixed subrepresentation.) (Note also that a non-zero $G$-equivariant map out of $\rho$ is automatically injective, because $\rho$ is irreducible.) |
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