A Tower Construction for the Radical in Brauer′s Centralizer Algebras Original Research Article

In this paper we study the structure of the Brauer centralizer algebras in the case that the multiplication constant x is a rational integer. Several authors have studied the structure of these algebras for generic values of x. In particular, Hans Wenzl showed that the Brauer algebras can be obtained from Jones′ Basic Construction and he used that fact to prove that the Brauer algebras are semisimple when x is not a rational integer. The Tower Construction is a method to study towers of semisimple algebras. Hence it does not apply in our case where the algebras in the tower eventually have radicals. Our first step in this paper is to modify the Tower Construction so that it does a simultaneous Tower Construction of the radicals and the semisimple quotients of a tower of algebras. The rest of the paper is devoted to describing these constructions explicitly in the Brauer algebra case. One surprising corollary of this method is a connection between two seemingly distinct criteria for the simplicity of certain subrings of the Brauer algebras. It is possible to identify explicitly certain subrings of the Brauer algebras which are the matrix rings corresponding to irreducible representations in the semisimple case. In previous work, these authors gave a combinatorial condition for simplicity of these individual matrix rings when x is a rational integer. The Tower Construction gives a second algebraic condition for simplicity. It is difficult to see why these two conditions are equivalent. However, the methods used in this paper make that clear.