Trends and lines of development in scheme theory

Abstract

The concept of an association scheme is one of those mathematical concepts which were utilized as technical tools in various different mathematical areas for a long time before becoming the subject of a theory in their own right. The significance of symmetric schemes, for instance, in the design of (statistical) experiments was recognized as early as the first half of the last century. Coding theory has been associated with commutative schemes for more than three decades, and polynomial schemes have provided the language in which major topics in algebraic graph theory are communicated for about twenty years. The notion of a scheme itself, however–a notion which, if considered in its full generality, generalizes not only the notion of a group but also the notion of a Moore geometry and that of a building in the sense of Jacques Tits–has been considered as the subject of an abstract theory in itself only relatively recently. It is the purpose of this article to reflect on the lines of development, the Entwicklungslinien, along which abstract scheme theory has been developed so far and along which scheme theory might be developed in the future. The emphasis will be not so much on completeness as on an attempt to show exemplarily how naturally and organically the structure theory of association schemes arises from certain aspects in group theory.