#### Vol. 7, No. 5, 2013

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Quantized mixed tensor space and Schur–Weyl duality

### Richard Dipper, Stephen Doty and Friederike Stoll

Vol. 7 (2013), No. 5, 1121–1146
##### Abstract

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

##### Keywords
Schur–Weyl duality, walled Brauer algebra, mixed tensor space, rational $q$-Schur algebra
##### Mathematical Subject Classification 2010
Primary: 33D80
Secondary: 16D20, 16S30, 17B37, 20C08