Theory of Association Schemes

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The present text is an introduction to the theory of association schemes. We start with the de?nition of an association scheme (or a scheme as we shall say brie?y), and in order to do so we ?x a set and call it X. We write 1 to denote the set of all pairs (x,x) with x? X. For each subset X ? r of the cartesian product X×X, we de?ne r to be the set of all pairs (y,z) with (z,y)? r.For x an element of X and r a subset of X× X, we shall denote by xr the set of all elements y in X with (x,y)? r. Let us ?x a partition S of X×X with?? / S and 1 ? S, and let us assume X ? that s ? S for each element s in S. The set S is called a scheme on X if, for any three elements p, q,and r in S, there exists a cardinal number a such pqr ? that|yp?zq| = a for any two elements y in X and z in yr. pqr The notion of a scheme generalizes naturally the notion of a group, and we shall base all our considerations on this observation. Let us, therefore, brie?y look at the relationship between groups and schemes.
Conceptual approach to association schemes is emphasized First rigorous treatment of the structure theory of schemes. Schemes are considered not necessarily to be commutative or finite. As a byproduct, the theory covers (disguised as ‘Coxeter schemes’) the theory of buildings in the sense of Jacques Tits Recent results such as the generalization of Sylow’s group-theoretical theorems to scheme theory and the characterization of Glauberman’s Z*-involutions in terms of scheme theory appear for the first time in book form

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Paul-Hermann Zieschang

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Arithmetic Morphism algebra association scheme building group proof theorem combinatorics

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