# Reference for quantum Schur-Weyl duality

I am trying to prove a version of quantum Schur-Weyl duality. I hope to be able to generalize the proof of the Schur-Weyl duality between $U_q(\mathfrak{gl}_n)$ and the Hecke algebra $H_r$. So I am looking for a good reference for this with a careful proof. It would also be nice to see a proof that uses the quantum coordinate ring of $GL_n$ instead of the enveloping algebra (and therefore is phrased in terms of decomposing $V^{\otimes r}$ as a left-comodule for this coordinate ring and a right $H_r$-module).

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This goes back to Jimbo I think. A reference is: "A q-difference analogue of $U(\mathfrak g)$, Hecke algebra and the Yang-Baxter equation'', Lett. Math. Phys. 11 (1986).

It has been much studied though, so there are lots of subsequent papers, some of which might be closer to what you are looking for? For example this paper studies an analogue of Schur-Weyl duality for "walled Brauer algebras", and this paper studies a two-parameter version.

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Hi Jonah - Section 4 of this paper deduces a quantum Schur-Weyl duality from quantum GL(n)-GL(m) Howe duality. But this doesn't use the quantum coordinate ring.

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Georgia Benkart and Sarah Witherspoon worked on a 2-parameter generalization, but also in quantized enveloping algebra language. See math/0108038.

There is also wide available work on various q-Schur algebras. It is hard to find what is not known as far as various basic analogues here are concerned.

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This is treated in section 8.6 of the book "Quantum Groups and Their Representations" by Klimyk and Schmudgen, although also not using the quantum coordinate algebra.

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Too late maybe but still - "Noncommutative symmetric functions V: a degenerate version of U_q(gl_N)" by Krob and Thibon has the quantum coordinate ring version very carefully (I think) written. It is aimed at the $q=0$ case but the generic $q$ is treated in detail too.

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