Quantized mixed tensor space and Schur–Weyl duality

Dipper, Richard; Doty, Stephen; Stoll, Friederike

doi:10.2140/ant.2013.7.1121

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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

DOI: 10.2140/ant.2013.7.1121

Abstract

Let $R$ be a commutative ring with $1$ and $q$ an invertible element of $R$ . The (specialized) quantum group $U = U_{q} ({g l}_{n})$ over $R$ of the general linear group acts on mixed tensor space $V^{\otimes r} \otimes {V^{*}}^{\otimes s}$ , 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 ${End}_{R} (V^{\otimes r} \otimes {V^{*}}^{\otimes s})$ 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 ${End}_{B_{r, s}^{n} (q)} (V^{\otimes r} \otimes {V^{*}}^{\otimes s}) = S_{q} (n; r, s)$ . By a previous result, the image of $B_{r, s}^{n} (q)$ in ${End}_{R} (V^{\otimes r} \otimes {V^{*}}^{\otimes s})$ is ${End}_{U} (V^{\otimes r} \otimes {V^{*}}^{\otimes s})$ . Thus, a mixed tensor space as $(U, B_{r, s}^{n} (q))$ -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

Milestones

Received: 11 November 2011

Revised: 12 April 2012

Accepted: 20 June 2012

Published: 6 September 2013

Authors

Richard Dipper
Institut für Algebra und Zahlentheorie Universität Stuttgart Pfaffenwaldring 57 70569 Stuttgart Germany

Stephen Doty
Department of Mathematics and Statistics Loyola University Chicago 1023 West Sheridan Road Chicago, IL 60660 United States

Friederike Stoll
Institut für Algebra und Zahlentheorie Universität Stuttgart Pfaffenwaldring 57 70569 Stuttgart Germany

Vol. 7, No. 5, 2013