Trends and lines of development in scheme theory

AbstractThe 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

Characters of association schemes containing a strongly normal closed subset of prime index

ArticlePROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. 135(9): 2683-2687 (2007)journal articl

Representation theory of imprimitive of non-commutative association schemes of order

In this note, we investigate association schemes of order 6. We prove that non-normal closed subsets of such schemes have order 2 and that normal closed subsets of non-commutative schemes have order 2 or 3. After that, we investigate more closely schemes of order 6 which possess non-normal closed subsets and non-commutative schemes of order 6 which possess a symmetric normal closed subset of order 3. (Non-commutative schemes of order 6 which possess a non-symmetric normal closed subset of order 3 or a normal closed subset of order 2 will be investigated in a forthcoming article.) In both cases, we explicitly give all irreducible (complex) representations of such schemes. Among other structural consequences we obtain that association schemes of order 6 are Coxeter schemes if they have two di erent non-normal closed subsets and that they are semidirect products if they possess a normal and a non-normal closed subset or a thin non-normal closed subset