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In my sourcebook for Lie algebra, I have a list of suggested problems to solve to familiarize myself with the subjects. One of those problems is:

Using Characters, prove the following formula $$V_{\lambda}\otimes V_{\mu}=\bigoplus^{\lambda+\mu}_{\nu=|\lambda-\mu|}V_{\nu}.$$

I am quite lost on how to solve this, the only idea that I may have is that maybe Weyl’s character formula can help, as this problem is right after that result. Any help would be really appreciated as I am not sure where to even begin.

AdrinMI49
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    What is $V_{\lambda}$ ? – Tuvasbien Dec 12 '23 at 01:19
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    Without context, I'm guessing you are talking about $\mathfrak{sl}_2$-reps. If so, https://math.stackexchange.com/questions/3384483/tensor-product-of-two-irreducible-mathfrak-sl-2-modules is your answer and this would be a duplicate. (You are correct that computing the character solves the problem). – MPos Dec 12 '23 at 12:26
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    Agree with @MPos. This must be about the tensor products of irreducible representations of $\mathfrak{sl}_2$ only. For other (semi)simple Lie algebras the sum is similar, but more complicated. Some sources call this the Clebsch-Gordan formula, but physicists in particular seem to use that pair of names when describing the formula for explicit vectors belonging to each irreducible summand. – Jyrki Lahtonen Dec 12 '23 at 12:46
  • I think as well that is the answer, sorry i disapeared. Unfortunately the problem in my book has exactly that statement, so I was as super lost. – AdrinMI49 Dec 12 '23 at 17:04

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