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Proving finite presentation

I am following the notation in the comments. I do not have privilege to make a comment, do I should write and answer. Also from the comments I understand that you are working just with commutative ...
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Clebsch–Gordan decomposition formula for algebraic groups

Let $G$ denote the group, and suppose one has an enumeration of its irreducible representations $V_{\lambda}$ by some combinatorial objects $\lambda$, like integers for $SU_2$ (or finite dimensional ...

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