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

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