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For a theorem that appears in every other book on representation theory, Schur-Weyl duality seems to have a shortage of actually distinct proofs. The argument (as given, e.g., in §4.18 and §4.19 of Pavel Etingof et al, Introduction to representation theory, arXiv:0901.0827v5) proceeds, roughly, as follows: [EDIT: The proof outlined in the following is neither the simplest nor the slickest version of the standard argument. The Etingof-et-al text does it in a much clearer way, by factoring out some of the semisimple-modules arguments into a general lemma. As pointed out by commenters, David Speyer and Mark Wildon (in MO question #90094) have further elementarized the argument, but their versions are still not as lightweight as I'd like them to be (e.g., they still use Schur's lemma, requiring proof of absolute irreducibility).]

For a theorem that appears in every other book on representation theory, Schur-Weyl duality seems to have a shortage of actually distinct proofs. The argument (as given, e.g., in §4.18 and §4.19 of Pavel Etingof et al, Introduction to representation theory, arXiv:0901.0827v5) proceeds, roughly, as follows: [EDIT: The proof outlined in the following is neither the simplest nor the slickest version of the standard argument. The Etingof-et-al text does it in a much clearer way, by factoring out some of the semisimple-modules arguments into a general lemma. As pointed out by commenters, David Speyer and Mark Wildon (in MO question #90094) have further elementarized the argument, but their versions are still not as lightweight as I'd like them to be (e.g., they still use Schur's lemma, requiring proof of absolute irreducibility).]

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\right) \right] $ on the tensor power $V^{\otimes n}$ are each other's centralizers) for a field  $\mathbb{K}$ of characteristic $0$ proven constructively?

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\right) \right] $ on the tensor power $V^{\otimes n}$ are each other's centralizers) for a field  $\mathbb{K}$ of characteristic $0$ proven constructively?

UPDATE: In comments to this post, Frieder Ladisch has alerted me to the fact that the double centralizer theorem (or, rather, the part of the double centralizer theorem that is relevant to the proof of part b)) can be proven constructively (provided that the input is sufficiently explicit). And now I am seeing that essentially his proof appears in Section 11.1 of Jan Draisma and Dion Gijswijt, Invariant Theory with Applications. (Jan: I took the freedom to guess the URL of the PDF file, seeing that the hyperlink was broken due to an incorrect relative path. If you actually don't want these notes to be linked, please let me know!) Some parts of their argument need to be slightly modified to ensure constructivity: The use of continuity in the proof of Theorem 11.1.1 should be replaced by a straightforward argument using Zariski density. The group $H$ in Theorem 11.1.2 should be required to be finite. The vector space $W$ in Theorem 11.1.2 should be required to be finite-dimensional. The requirement in Theorem 11.1.2 that the representation $\lambda$ be completely reducible should be replaced by a requirement that $\left|H\right|$ is invertible in the ground field. The direct complement $U$ of $M$ in the proof of Theorem 11.1.2 should be constructed using Maschke's theorem, which has a well-known proof relying merely on linear algebra (viz., the existence of a complement of an explicitly-defined subspace of a finite-dimensional vector space).

Of course, this beautiful argument still "feels inexplicit" in the sense that it uses some representation-theoretical ideas. But the worst offenders (Artin-Wedderburn theory, passing to algebraic closure, analysis/geometry etc.) are gone. Had I known this argument in advance, I wouldn't have asked this question. Nevertheless, I am leaving this question open, since I have yet to digest various other answers, some of which appear to lead to more general proofs, maybe even in positive characteristic (for whatever parts of Schur-Weyl duality hold there).

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\right) \right] $ on the tensor power $V^{\otimes n}$ centralize each other) for a field $\mathbb{K}$ of characteristic $0$ proven constructively?

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\right) \right] $ on the tensor power $V^{\otimes n}$ are each other's centralizers) for a field $\mathbb{K}$ of characteristic $0$ proven constructively?