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$V$ is an $m$ dimensional vector space having a structure of $sl_2(\mathbb{C})$-module, where $sl_2(\mathbb{C})$ is the Lie algebra of the Lie group $SL_2(\mathbb{C})$. The symmetric group $S_n$ acts on the tensor product $V^{\otimes n}$.

What does Schur-Weyl duality say in this case?

What is the irreducible decomposition of $V^{\otimes n}$?

If we have $S_n$ irreducible decomposition can we get $sl_2(\mathbb{C})$ decomposition and vice versa?

I would be very grateful if someone could give a detailed answer. Thanking you in advance.

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    The classical [Schur-Weyl duality](http://en.wikipedia.org/wiki/Schur-Weyl_duality) only applies for the actions of $GL(V)$ diagonally on $V^{\otimes}$ and $S_n$ by permutation of the factors. If you just have an $sl_2(\mathbb{C})$ action on $V$, then the first thing to ask is how this decomposes into irreducibles. And even if $V$ is an irreducible $sl_2(\mathbb{C})$-module, you are not in the situation of the Schur-Weyl duality; indeed the tensor product typically has more irreducible components than there are partitions of $n$.2012-06-08

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