Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 5 (2013), 1121-1146.
Quantized mixed tensor space and Schur–Weyl duality
Let be a commutative ring with and an invertible element of . The (specialized) quantum group over of the general linear group acts on mixed tensor space , where denotes the natural -module , and are nonnegative integers and is the dual -module to . The image of in is called the rational -Schur algebra . We construct a bideterminant basis of . There is an action of a -deformation of the walled Brauer algebra on mixed tensor space centralizing the action of . We show that . By a previous result, the image of in is . Thus, a mixed tensor space as -bimodule satisfies Schur–Weyl duality.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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. https://projecteuclid.org/euclid.ant/1513730007