Algebra & Number Theory

Quantized mixed tensor space and Schur–Weyl duality

Richard Dipper, Stephen Doty, and Friederike Stoll

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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.

Article information

Algebra Number Theory, Volume 7, Number 5 (2013), 1121-1146.

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

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Primary: 33D80: Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke algebras, and related topics
Secondary: 16D20: Bimodules 16S30: Universal enveloping algebras of Lie algebras [See mainly 17B35] 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 20C08: Hecke algebras and their representations

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


Dipper, Richard; Doty, Stephen; Stoll, Friederike. Quantized mixed tensor space and Schur–Weyl duality. Algebra Number Theory 7 (2013), no. 5, 1121--1146. doi:10.2140/ant.2013.7.1121.

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