paper

Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

arXiv:math/0503545

Abstract

In this paper we prove Schur-Weyl duality between the symplectic group and Brauer algebra over an arbitrary infinite field $K$ . We show that the natural homomorphism from the Brauer algebra $B_{n} (- 2 m)$ to the endomorphism algebra of tensor space $(K^{2 m})^{\otimes n}$ as a module over the symplectic similitude group $GS p_{2 m} (K)$ (or equivalently, as a module over the symplectic group $S p_{2 m} (K)$ ) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $GS p_{2 m} (K)$ to the endomorphism algebra of $(K^{2 m})^{\otimes n}$ as a module over $B_{n} (- 2 m)$ , is derived as an easy consequence of S.~Oehms' results [S. Oehms, J. Algebra (1) 244 (2001), 19--44].

27 pages