I had a question which is slightly more general than this one on mathoverflow: I am looking for an explicit description of the isomorphism $\mathbb S_\nu(V\otimes W) \cong \bigoplus C_{\lambda\mu\nu} \mathbb S_\lambda V\otimes \mathbb S_\mu W$ from Exercise 6.11 in Fulton & Harris. In said question, the author asks whether $${\textstyle\bigwedge}^p(V\otimes W) \cong \bigoplus\nolimits_{\substack{\lambda\vdash p\\\\\ell(\lambda)\le n\\\\\lambda_1\le m }} \mathbb{S}_\lambda V \otimes \mathbb{S}_{\bar\lambda}W$$

is given by

$$ (v_1\otimes w_1)\wedge\ldots\wedge(v_p\otimes w_p) \longmapsto \sum\nolimits_{\substack{\lambda\vdash p\\\\\ell(\lambda)\le n\\\\\lambda_1\le m }} c_\lambda(v_1\otimes\ldots\otimes v_p) \otimes c_{\bar\lambda}(w_1\otimes\ldots\otimes w_p). $$

The answer is affirmative and in the comments it is said that this is due to Schur-Weyl duality. I frankly don't understand that argument: I do not see why the map is well-defined. One would have to show that simultaneous permutation of the $v_i$ and the $w_i$ with some $\pi\in S_p$ introduces a factor of $\mathrm{sgn}(\pi)$ on the right, and this is not at all clear to me. Can someone explain?