The Quantized Walled Brauer Algebra and Mixed Tensor Space |
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Authors: | R Dipper S Doty F Stoll |
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Institution: | 1. Institut für Algebra und Zahlentheorie, Universit?t Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany 2. Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, IL, 60660, USA
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Abstract: | In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (1994). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring R of rank independent of R. As an application, we prove one side of Schur–Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the R-endomorphism ring of mixed tensor space is, for all choices of R and the parameter q, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra U of the general linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the U-invariants in the ring of R-linear endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$ . |
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