Question #23893

Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3),
on which G acts by permuting the ei’s. Is G a completely reducible linear group?

Expert's answer

Since (123) does not act triviallyon *V *, the representation homomorphism *ϕ *: *G → *GL(*V *) must be injective.Therefore, *ϕ*realizes *G *as alinear group in GL(*V *). It is easy to see that *V*_{o} = *k*(*e*1*− e*2) is a *kG – *module affording the sign representation (of *G*),with *V/V*_{o} affording the trivial representation, so* kGV *isindecomposable. Therefore, *G **⊆*GL(*V *) is not completelyreducible.

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