The Schur-Weyl duality between quantum group of type \(A\) and Hecke algebra (Q2767400)

\textit{M. Jimbo} [Lett. Math. Phys. 11, 247-252 (1986; Zbl 0602.17005)] pointed out the duality between quantum group of type A and the Hecke algebra when \(q\) is not a root of unity, as a deformation of the Schur-Weyl duality between the classical groups of type A and the group algebra \(CS_{n}\) of the symmetric group \(S_{n}\). The following formulae were claimed by Jimbo: NEWLINE\[NEWLINE H(q,n)= \text{End}_{U_{q}(SL_{m})}(V_{m}^{\otimes{n}}), \qquad U_{q}(SL_{m})= \text{End}_{H(q,n)}(V_{m}^{\otimes{n}}) NEWLINE\]NEWLINE without proof, where \(H(q,n)\) is a Hecke algebra, \(U_{q}(SL_{m})\) a quantum group, \(V_{m}\) the standard representation of \(U_{q}(SL_{m})\). In this paper, the authors give a complete proof of these claims. The method they use is to decompose the tensor powers \(V_{m}^{\otimes{n}}\) into the direct sum of irreducible representations of \(U_{q}(SL_{m})\) through the action of Hecke algebras. They also show the duality between \(U_{q}(SL_{\infty})\) and \(H(q,n)\) on the Hilbert space \(H^{\otimes{n}}\) when \(q\) is a real number.