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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 $X=G/H$.

edited Sep 7 2010 at 17:50

Jim Humphreys
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asked Sep 7 2010 at 16:01

Klim Efremenko
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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 2010 at 17:55

Emerton · Accepted Answer · 2010-09-07 16:28:07Z UTC

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.)

Embedding into Permutation Representation

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