MATHEMATICS RESEARCH AREAS

DANE FLANNERY is interested in interactions between combinatorial design theory and cohomology of finite groups. A cocyclic matrix over a finite group G displays explicitly the action of a 2-cocycle defined over G. An Hadamard matrix is a square (+1,-1)-matrix whose distinct rows are pairwise orthogonal. While it is unknown whether Hadamard matrices exist at every possible order, many examples (including several infinite families) of cocyclic Hadamard matrices have been catalogued, using the equivalence between these objects, relative difference sets, and group divisible designs. There are viable computational methods for constructing cocyclic matrices. However, determining cocyclic Hadamard matrices is a problem that depends exponentially on the size of G. A particular action of G on the abelian group of its 2-cocycles with coefficients in a cyclic group of prime order p leads naturally to a matrix group representation of G over the field of size p. Ongoing investigations are concerned with how the theory of such representations may be used to reduce the complexity of the general problem of searching for cocyclic Hadamard matrices.

GOETZ PFEIFFER is interested in finite groups, representation theory, computational group theory, algebraic combinatorics and semigroups. He has participated in the development of the GAP system for computational group theory. He is one of the authors of the CHEVIE package for generic character tables of finite groups of Lie type and associated structures like Weyl groups and Hecke algebras. He has co-authored a book about the representation theory of finite Coxeter groups and the associated Iwahori-Hecke algebras. He has developed a GAP package for structural investigations of finite monoids. Currently, he is working on algorithms for descent algebras of finite Coxeter groups.