Brauer Algebras and Centralizer Algebras for SO(2n, C)~1

摘要

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~(direct X 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))-bar its largest associative quotient. We then show that when k <= 2n, the semisimple quotient of (D_k(2n))-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.