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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?2012-02-16
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    I don't see why it solves the exercise; sorry for not being clear.2012-02-16
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    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 $V$?2012-02-16
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    Yes, by Schur; so since both are irreducible weshould have that the dim is $1$ if $U = V$ or $0$ otherwise...2012-02-16
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    I am not requiring that $V$ is irreducible (since for the application above it isn't).2012-02-16
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    In FH it says that V is also irreducible, but anyway, do continue :)2012-02-16
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    That's just notation. Okay, let me use a different letter. If $U$ is irreducible and $X$ is just some representation, can you see why $\dim \text{ Hom}(X, U)$ and $\dim \text{ Hom}(U, X)$ both count the multiplicity of $U$ in $X$, and can you see why the result follows from this?2012-02-16
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    I see why that is true, but I don't see why the result follows, sorry2012-02-16
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    Okay, apply the result to $\text{Hom}(V \otimes W, U)$ and then apply it again to $\text{Hom}(W, V^{\ast} \otimes U)$. What do you conclude?2012-02-16
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    Ah, I got that :D thanks! I was thinking about the last part2012-02-16
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    I got it, sorry (too tired). Thanks a lot for your help.2012-02-16

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