On Maximal Closed Subsets in Association Schemes

✍ Scribed by Paul-Hermann Zieschang

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 274 KB
• Volume: 80
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2792

✦ Synopsis

We offer a sufficient condition for a closed subset in an association scheme to be maximal. The result generalizes naturally the well-known (group-theoretical) fact that the one-point-stabilizer of a flag transitive automorphism group of a 2-design with *=1 must be a maximal subgroup. In finite group theory, there exist a lot of (important) sufficient conditions for a subgroup to be maximal. In the theory of association schemes, the condition which we offer here seems to be the first one which guarantees that a closed subset of the set of relations of an association scheme is maximal.

1997 Academic Press

The starting point for the classification [2] of flag transitive automorphism groups of finite 2-designs with *=1 is an elementary observation due to Higman and McLaughlin. It is the well-known fact that a group which acts flag transitively on a 2-design with *=1 must act primitively on the set of points of the design in question; see [5, Proposition 3]. Various variants on this result were found later, either from a combinatorial point of view (cf., e.g., [3, 2.3.7(a)]) or from a group-theoretical point of view (cf., e.g., [7]).

In [8, (1.12)], it was shown that the class of groups can be viewed naturally as a distinguished class of association schemes. 1 Moreover, when going over from group theory to the theory of schemes, many basic concepts (of group theory) are generalized naturally to well-known algebraic or geometric concepts. (For example, group algebras are generalized to Hecke algebras, and Coxeter groups to Tits buildings; see [8,9].) The purpose of the present note is to show that also the above-mentioned result of Higman and McLaughlin has a natural generalization in scheme theory.