Brauer Algebras And Centralizer Algebras For SO(2n,C)

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Journal Of Algebra


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.