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a representation which is not completely reducible
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(Example)
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If $G$ is a finite group, and $k$ is a field whose characteristic does divide the order of the group, then Maschke's theorem fails. For example let $V$ be the regular representation of $G$ , which can be thought of as functions from $G$ to $k$ , with the $G$ action $g\cdot\vp(g')=\vp(g^{-1}g')$ . Then this representation is not completely reducible.
There is an obvious trivial subrepresentation $W$ of $V$ , consisting of the constant functions. I claim that there is no complementary invariant subspace to this one. If $W'$ is such a subspace, then there is a homomorphism $\vp:V\to V/W'\cong k$ . Now consider the characteristic function of the identity $e\in G$
and $\ell=\vp(\delta_e)$ in $V/W'$ . This is not zero since $\delta$ generates the representation $V$ . By $G$ -equivarience, $\vp(\delta_g)=\ell$ for all $g\in G$ . Since $$\eta=\sum_{g\in G}\eta(g)\delta_g$$ for all $\eta\in V$ , $$W'=\vp(\eta)=\ell\left(\sum_{g\in G}\eta(g)\right).$$ Thus, $$\ker\vp=\{\eta\in V|\sum_{\in G}\eta(g)=0\}.$$ But since the characteristic of the field $k$ divides the order of $G$ , $W\leq W'$ , and thus could not possibly be complementary to it.
For example, if $G=C_2=\{e,f\}$ then the invariant subspace of $V$ is spanned by $e+f$ . For characteristics other than $2$ , $e-f$ spans a complementary subspace, but over characteristic 2, these elements are the same.
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"a representation which is not completely reducible" is owned by bwebste.
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Cross-references: complementary subspace, spans, spanned by, generates, identity, characteristic function, homomorphism, subspace, invariant subspace, complementary, constant functions, subrepresentation, obvious, completely reducible, representation, action, functions, regular representation, Maschke's theorem, group, order, divide, characteristic, field, finite group
This is version 2 of a representation which is not completely reducible, born on 2003-03-24, modified 2005-03-10.
Object id is 4122, canonical name is ARepresentationWhichIsNotCompletelyReducible.
Accessed 2361 times total.
Classification:
| AMS MSC: | 20C15 (Group theory and generalizations :: Representation theory of groups :: Ordinary representations and characters) |
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Pending Errata and Addenda
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