Characters of Brauer's Centralizer Algebras

Brauer's centralizer algebras are finite dimensional algebras with a distinguished basis. Each Brauer centralizer algebra contains the group algebra of a symmetric group as a subalgebra and the distinguished basis of the Brauer algebra contains the permutations as a subset. In view of this containment it is desirable to generalize as many known facts concerning the group algebra of the symmetric group to the Brauer algebras as possible. This paper studies the irreducible characters of the Brauer algebras in view of the distinguished basis. In particular we define an analogue of conjugacy classes, and derive Frobenius formulas for the characters of the Brauer algebras. Using the Frobenius formulas we derive formulas for the irreducible character of the Brauer algebras in terms of the irreducible characters of the symmetric groups and give a combinatorial rule for computing these irreducible characters.