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2013 Quantized mixed tensor space and Schur–Weyl duality
Richard Dipper, Stephen Doty, Friederike Stoll
Algebra Number Theory 7(5): 1121-1146 (2013). DOI: 10.2140/ant.2013.7.1121


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 VrVs, 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(VrVs) 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)(VrVs)=Sq(n;r,s). By a previous result, the image of Br,sn(q) in EndR(VrVs) is EndU(VrVs). Thus, a mixed tensor space as (U,Br,sn(q))-bimodule satisfies Schur–Weyl duality.


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Richard Dipper. Stephen Doty. Friederike Stoll. "Quantized mixed tensor space and Schur–Weyl duality." Algebra Number Theory 7 (5) 1121 - 1146, 2013.


Received: 11 November 2011; Revised: 12 April 2012; Accepted: 20 June 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1290.17012
MathSciNet: MR3101074
Digital Object Identifier: 10.2140/ant.2013.7.1121

Primary: 33D80
Secondary: 16D20 , 16S30 , 17B37 , 20C08

Keywords: mixed tensor space , rational $q$-Schur algebra , Schur–Weyl duality , walled Brauer algebra

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 5 • 2013
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