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 = Uq(gln) over R of the general linear group acts on mixed tensor space V r V s, where V denotes the natural U-module Rn, r and s are nonnegative integers and V is the dual U-module to V . The image of U in EndR(V r V s) is called the rational q-Schur algebra Sq(n;r,s). We construct a bideterminant basis of Sq(n;r,s). There is an action of a q-deformation Br,sn(q) of the walled Brauer algebra on mixed tensor space centralizing the action of U. We show that EndBr,sn(q)(V r V s) = Sq(n;r,s). By a previous result, the image of Br,sn(q) in EndR(V r V s) is EndU(V r V s). Thus, a mixed tensor space as (U,Br,sn(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