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Again something from Fulton and Harris I'm having trouble with:

Exercise 2.33 (c). If $U$, $V$, and $W$ Are irreducible representations, show that $U$ appears in $V \otimes W$ if and only if $W$ occurs in $V^{*} \otimes U$. Deduce that this cannot occur unless $\dim U \geq \dim W / \dim V$.

The hint suggests to look at the fact that $\mathrm{Hom}_{G}(V \otimes W, U) = \mathrm{Hom}_{G}(W, V^{*} \otimes U)$...

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What trouble are you having with the hint? Do you not see why that statement is true or do you not see why it solves the exercise? – Qiaochu Yuan Feb 16 '12 at 4:23
I don't see why it solves the exercise; sorry for not being clear. – Anna Feb 16 '12 at 4:25
If $U$ is irreducible, can you see why $\dim \text{ Hom}(V, U)$ and $\dim \text{ Hom}(U, V)$ both count the multiplicity of $U$ in $V$? – Qiaochu Yuan Feb 16 '12 at 4:26
Yes, by Schur; so since both are irreducible weshould have that the dim is $1$ if $U = V$ or $0$ otherwise... – Anna Feb 16 '12 at 4:32
I am not requiring that $V$ is irreducible (since for the application above it isn't). – Qiaochu Yuan Feb 16 '12 at 4:36

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