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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Brauer algebras, symplectic Schur algebras and Schur-Weyl duality


Authors: Richard Dipper, Stephen Doty and Jun Hu
Journal: Trans. Amer. Math. Soc. 360 (2008), 189-213
MSC (2000): Primary 16G99
Published electronically: August 16, 2007
MathSciNet review: 2342000
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Abstract: In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field $ K$. We show that the natural homomorphism from the Brauer algebra $ B_{n}(-2m)$ to the endomorphism algebra of the tensor space $ (K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $ GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $ Sp_{2m}(K)$) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $ GSp_{2m}(K)$ to the endomorphism algebra of $ (K^{2m})^{\otimes n}$ as a module over $ B_{n}(-2m)$, is derived as an easy consequence of S. Oehms's results [S. Oehms, J. Algebra (1) 244 (2001), 19-44].


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Additional Information

Richard Dipper
Affiliation: Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany
Email: Richard.Dipper@mathematik.uni-stuttgart.de

Stephen Doty
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 6525 North Sheridan Road, Chicago, Illinois 60626
Email: doty@math.luc.edu

Jun Hu
Affiliation: Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
Email: junhu303@yahoo.com.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04179-7
PII: S 0002-9947(07)04179-7
Received by editor(s): April 8, 2005
Received by editor(s) in revised form: September 27, 2005
Published electronically: August 16, 2007
Additional Notes: The first author received support from DOD grant MDA904-03-1-00.
The second author gratefully acknowledges support from DFG Project No. DI 531/5-2
The third author was supported by the Alexander von Humboldt Foundation and the National Natural Science Foundation of China (Project 10401005) and the Program NCET
Article copyright: © Copyright 2007 American Mathematical Society



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