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 Ek(2n)=EndSO(2n)(V⊗k) for V=C2n is a set of diagrams. To each diagram d, Brauer associated a linear transformation Φ(d) in Ek(2n) and showed that Ek(2n) is spanned by the transformations Φ(d). In this paper, we first define a product on Dk(2n), the C-linear span of the diagrams. Under this product, Dk(2n) becomes an algebra, and Φ extends to an algebra epimorphism. Since Dk(2n) is not associative, we denote by Dk(2n) its largest associative quotient. We then show that when k≤2n, the semisimple quotient of Dk(2n) is equal to Ek(2n). Next, we prove some facts about the representation theory of Ek(2n). We compute the dimensions of the irreducible Ek(2n)-modules and give some branching rules.