# Reference Request: Jimbo's Proof of Quantum Schur-Weyl Duality

In his seminal 1986 paper "A $$q$$-analogue of $$U(\mathfrak{gl}(N+1))$$, Hecke Algebra, and the Yang-Baxter Equation", Jimbo asserted (Proposition 3) that the quantum group associated to $$\mathfrak{gl}_n$$ and the Hecke algebra associated to $$S_n$$ are in Schur-Weyl duality. But the paper doesn't have the proofs, and Jimbo says in the introduction "Details will appear elsewhere".

Did the details ever actually appear? I haven't been able to find them anywhere, or even a citation to a longer paper.

I'm interested for curiosity's sake in Jimbo's original proof, but more broadly, I mostly just want a (hopefully clear) full proof of this result. Every source I've found either sketches the proof or does it for a related case.

• Well, at least a reference: Chari, Pressley, A guide to quantum groups, see Sect 10.2, esp Thm 10.2.5 (i), Prop 10.2.6 and the bibliographical notes on p337. I'm not sure if it meets your criteria, but it's at least a start. See also the references at mathoverflow.net/q/15392. – Jules Lamers May 24 at 2:17
• Thanks! Chari and Pressley is a great source, but as far as I can tell, doesn't give a proof. Am I missing something? I'm going to take a look as some of the links in the other post though! – Andy Hardt May 24 at 4:30
• Yeah, I think you're right. The idea is that the $\check{R}$-matrix, i.e. the permutation times the R-matrix, coincides (possibly up to a factor: $\epsilon$ for eq (20) on p335 of Chari--Pressley) with the Hecke generator. The 'RTT relation', i.e. R intertwines the ordinary and opposite coproducts, thus implies that the Hecke-algebra action commutes with the representation of $U_q(\mathfrak{gl}_N)$ (which is defined via the repeated coproduct). – Jules Lamers May 24 at 4:43
• ...There's a subtlety though: that representation of the Hecke algebra is not faithful, but factors through a representation of the Temperley--Lieb algebra. This is worked out in detail in e.g. the BSc thesis of Weelinck, Representation Theory of the Temperley--Lieb algebra and its connections with the Hecke algebra (UvA, 2012) – Jules Lamers May 24 at 5:01