When trying to show that a representation $\rho:G\to GL(V)$ is not completely reducible (semisimple), is it enough to find a subspace $W\subset V$ which is $\rho(G)$-invariant and show that the complementary subspace is not $\rho(G)$-invariant?
Proving a representation is not semisimple
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linear-algebra
representation-theory
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7There is no "the" complementary subspace. You need to show that there is no complementary subspace which is invariant. – 2017-02-02