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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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Minor edits. I guess F is the underlying field. Note that the characteristic of the field and whether G is finite might influence the answer or the methods used. Permutation representations most often come up for finite groups. –  Jim Humphreys Sep 7 '10 at 17:55

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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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Note also that, if $\chi$ is the character of $\rho$, then $Hom_H(\rho, trivial)$ has dimension $1/|H| \sum_{h \in H} \chi(h)$. –  David Speyer Sep 7 '10 at 17:48
    
Sometimes I feel like I use this fact 5000 times a day. –  JSE Sep 7 '10 at 18:26

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