Combinatorial Representation Theory: Constructive and Integral

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Liebler This award supports an investigation into a variety of detailed research programs, including cyclic and non-abelian difference sets, Williamson matrices, finite geometries from special p-groups, cyclotomic association schemes and integral Hecke algebras. Although each is of substantial interest in its own right, further study of each of these topics will also contribute to the broader goal of development of an integral representation theory for association scheme-like structures that can be used constructively. This investigation is computation-intensive, relying heavily on the high-level computer software packages. This award also supports guest speakers at an interstate Algebraic Combinatorics Seminar. The biweekly seminar benefits students and faculty from universities along the front range of Colorado and Wyoming. This research is in the general area of Combinatorics. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. The broad goal of this research is to narrow the gap between the known examples and theoretical limits on certain types of highly symmetric discrete systems.