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I'm reading Introduction to Representation Theory by Pavel Etingof et al. and I want to do most of the exercises. But I must stop by the exercise on page 25. I can prove part (a) easily by direct proof and also by contradiction. I also can prove the first part of the hint: $V = E_{11} V \oplus E_{22}V \oplus \cdots \oplus E_{dd}V$, but than I don't know what to do to finish the exercise. Can someone help me?!

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closed as unclear what you're asking by Julian Kuelshammer, Normal Human, Batominovski, anomaly, user26857 Aug 17 at 7:53

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So which of the following that you were not able to do? (1) $\Phi_i$ is an isomorphism (2) $S(v)$ a subrep of $V$ (3) $S(v)$ isom to $k^d$ (4) $v\in S(v)$ (5) $V=S(v_1) \oplus \cdots \oplus S(v_k)$ with $v_1,\ldots, v_k$ basis of $E_11 V$. –  Aaron Dec 17 '12 at 21:42