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My multilinear algebra is pretty bad, so I just wanted to check if I my intuition is correct:

$$ Hom_{S_n}(V^{\otimes n}, V^{\otimes n}) \cong Sym^n(End(V)) $$

where V is a finite dimensional vector space and $S_d$ is the group of permutations of order $d$. Every linear map that commutes with permutation of the coordinates must be basically of the form $M\otimes \cdots \otimes M$. Am I off the mark?

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