Wedge-direct sums of table algebras and applications to association schemes, II

Abstract

Extensions of association schemes and table algebras have been studied in many papers in the last two decades. A remarkable operation (the Blau-construction) on two table algebras in [3] provides an important method to construct the extension of table algebras. The wedge-direct sum of a sequence of table algebras introduced in Part I (see [14]) is a useful tool for the study of structures and characterizations of table algebras. Using the wedge-direct sum of a sequence of table algebras, in Part I we gave a new perspective on the Blau-construction, and obtained a clear description of the structure of table algebras constructed by recursively applying the Blau-construction. p-table algebras (p-schemes) are an important class of table algebras (association schemes). In this part we will give the characterization and classification of a class of p-table algebras, by showing that they are isomorphic to the wedge-direct sum of a sequence of group algebras of finite p-groups. In order to determine whether two distinct sequences of table algebras yield the isomorphic wedge-direct sums or not, we will need to discuss the redundancy of the wedge-direct sum, and prove a Krull-Schmidt type theorem. Applications to association schemes will also be discussed.