Brauer algebras, symplectic Schur algebras and Schur-Weyl duality
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- by Richard Dipper, Stephen Doty and Jun Hu PDF
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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].References
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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
- MR Author ID: 59395
- ORCID: 0000-0003-3927-3009
- 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
- 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 - © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 189-213
- MSC (2000): Primary 16G99
- DOI: https://doi.org/10.1090/S0002-9947-07-04179-7
- MathSciNet review: 2342000