A combinatorial approach to the character theory of split metabelian groups

We call a finite group metabelian if its commutator subgroup is abelian. Thus a split metabelian group is a split extension of one abelian group by a second (possibly trivial) abelian group. In this paper we devise an algorithm for the construction of the character table of any split metabelian group G. The only requisites are that the reader be capable of deriving the character table for an abelian group (viz., that of the commutator subgroup [G, G] of G and that of the factor group G/[G, G]) and that he be able to identify the fusion of classes of [G, G] under the action of G/[G, G]. Thus the algorithm presupposes only a knowledge of that which is traditionally covered in a first year graduate course in algebra. Our verification of the algorithm relies heavily on a branch of algebraic combinatorics called association scheme theory. As the first eigenmatrix of an association scheme (in normalized form) generalizes the notion of a character table, this connection is hardly surprising. Still, it is reasonable to expect that the theory of association schemes may well provide a more natural point of departure for a variety of character theoretic problems and may aid greatly in their resolution.