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Let $G$ a Lie group and let $V$ a representations of $G$. Then we have the following representations are isomorphic: \begin{align} V \otimes V \cong S^2(V) \oplus \Lambda^2(V) \end{align}

I have no idea how to prove this fact. Any suggestions? Thanks in advance!

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    $\begingroup$ How about $v\otimes w\mapsto (v\otimes w + w\otimes v, v\otimes w - w\otimes v)$ ? $\endgroup$ – Max Feb 11 at 18:59
  • $\begingroup$ @Max It works, thank you!! $\endgroup$ – userr777 Feb 17 at 13:37

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