The Brauer algebra and the symplectic Schur algebra

Let k be an algebraically closed field of characteristic p > 0, let m, r be integers with m a parts per thousand yen 1, r a parts per thousand yen 0 and m a parts per thousand yen r and let S (0)(2m, r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. We also show that, in general, the block relations of the Brauer algebra and the symplectic and orthogonal Schur algebra are the same. Finally, we deduce from the previous results a new proof of the description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.