On homomorphisms of Brauer algebra modules in the non-semisimple case

At non-semisimple values, the structure of the radicals of Brauer's centralizer algebras is not well understood. In particular, the degeneracies of generic irreducibles have not been specifically classified by the modules that represent their intersection with the radical. There has also been no explanation of certain dimensional identities that hold for every example of these degeneracies that has been computed.; In order to achieve a better understanding of the radicals through combinatorial descriptions, the machinery of calculations with Brauer tableaux is developed more thoroughly than it has been before. This development takes the form of two sign-reversing involutions on Brauer polytabloids. Other combinatorial techniques are then used to show when homomorphisms into the radicals exist. This is done in the generic irreducibles to show their degeneracies.; The result is a complete classification of the degeneracies that can occur in generic irreducibles indexed by partitions of size four smaller than the number of vertices on the tops of the basis elements in the Brauer algebras. The technique also provides an alternative treatment of some cases that were already known for generic irreducibles, and suggests how to proceed further. The results also include a dimensional identity for modules associated with the radical in a non-trivial case, conjectured by P. Hanlon and D. Wales. This identity suggests a way to simplify calculations of the radicals even further.