Representation theory of finite groups and association schemes

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Groups are algebraic structures that can be used to understand symmetries of objects. The representation theory of groups is the area of algebra that makes it possible for us to represent these symmetries in a matrix form, which gives us a means to work with complicated structure using the familiar tools of linear algebra. A basic understanding of group representation theory is essential for an understanding of research being done now in computer science, chemistry, physics, and information theory. My work in group representation theory focuses on the possibilities that can arise when the entries of the matrices being used in the representations are restricted to the field of rational numbers. With this restriction, many complicated division algebra structures can be present that do not occur when the entries of the matrices are allowed to range freely over the real or complex numbers. One of the theoretical tools available for dealing with these division algebras is the Brauer group, something that appears in many places in mathematics. One of the goals of my proposal is look for a generalization of the Brauer group that could be of use in new approaches to some of the most important problems in the representation theory of finite groups. The other aspect of my proposal concerns association schemes, a combinatorial generalization of groups that was discovered to have useful applications in statistics and coding theory about 50 years ago. Another main goal of my proposal is to contribute to the development of a representation theory for association schemes along the lines of the representation theory for groups. Many of the things that we know for groups are open questions for association schemes. My students and I hope to be able to settle some of these issues in the course of this work, and open up the area of association schemes to appear future applications of representation theory.