Quantized mixed tensor space and Schur-Weyl duality

Let R be a commutative ring with 1 and q an invertible element of R. The (specialized) quantum group U = U-q(gl(n)) over R of the general linear group acts on mixed tensor space V-circle times r circle times V*(circle times s), where V denotes the natural U-module R-n, r and s are nonnegative integers and V* is the dual U-module to V. The image of U in End(R)(V-circle times r circle times V*(circle times s)) is called the rational q-Schur algebra S-q (n; r, s). We construct a bideterminant basis of S-q(n; r, s). There is an action of a q-deformation B-r,s(n)(q) of the walled Brauer algebra on mixed tensor space centralizing the action of U. We show that End(Br,s) (n(q))(V-circle times r circle times V*(circle times s)) = S-q(n; r, s). By a previous result, the image of B-r,s(n) (q) in End(R)(V-circle times r circle times V*(circle times s)) is End(U)(V-circle times r circle times V*(circle times s)). Thus, a mixed tensor space as (U, B-r,s(n)(q))-bimodule satisfies Schur-Weyl duality.