Theory of Association Schemes

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.