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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Brauer algebras, symplectic Schur algebras and Schur-Weyl duality
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by Richard Dipper, Stephen Doty and Jun Hu PDF
Trans. Amer. Math. Soc. 360 (2008), 189-213 Request permission

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
  • 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