Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

Details Brauer algebras, symplectic Schur algebras and Schur-Weyl duality Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

2005-03-24

arxiv.org/abs/math/0503545v2

Mathematics

Representation Theory

27 pages

Scientific paper

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 $B_n(-2m)$ to the endomorphism algebra of 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' results [S. Oehms, J. Algebra (1) 244 (2001), 19--44].

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