Brauer algebras and centralizer algebras for SO(2n, C)

In 1937, Richard Brauer identified the centralizer algebra of transformations commuting with the action of the complex special orthogonal groups SO(2n). Corresponding to the centralizer algebra E-k(2n) = End(SO(2n))(V(R)(k)) for V = C-2n is a set of diagrams. To each diagram d, Brauer associated a linear transformation Phi(d) in E-k(2n) and showed that E-k(2n) is spanned by the transformations Phi(d). In this paper, we first define a product on D-k(2n), the C-linear span of the diagrams. Under this product, D-k(2n) becomes an algebra, and Phi extends to an algebra epimorphism. Since D-k(2n) is not associative, we denote by <(D-k(2n))over bar> its largest associative quotient. We then show that when k less than or equal to 2n, the semisimple quotient of <(D-k(2n))over bar> is equal to E-k(2n). Next, we prove some facts about the representation theory of E-k(2n). We compute the dimensions of the irreducible E-k(2n)-modules and give some branching rules. (C) 2000 Academic Press.