Abstract

A diagram algebra is an algebra whose elements can be represented as linear combinations of diagrams. There are several common features of such algebras, allowing us to use similar methods in analysing their representation theory and obtaining similar results. In this talk I will focus on the Brauer and partition algebras, introduced by Brauer and Martin respectively. The representation theory of both of these over a field of characteristic zero is well understood. I will recall the definitions and give the block structure of both algebras in characteristic zero in terms of the action of a reflection group on the set of simple modules. I will then give a description of the blocks in positive characteristic by using the corresponding affine reflection group (for the partition algebra, this is joint work with C. Bowman and M. De Visscher). Finally I will show that by restricting our attention to specific families of these algebras, we can in fact obtain the entire decomposition matrix.