# Theory of Association Schemes

• Paul-Hermann Zieschang Book

Part of the Springer Monographs in Mathematics book series (SMM)

1. Front Matter
Pages I-XV
2. Pages 1-16
3. Pages 17-38
4. Pages 39-62
5. Pages 63-81
6. Pages 83-102
7. Pages 103-131
8. Pages 133-152
9. Pages 153-182
10. Pages 183-208
11. Pages 209-236
12. Pages 237-248
13. Pages 249-276
14. Back Matter
Pages 277-283

### Introduction

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.

### Keywords

Arithmetic Morphism algebra association scheme building group proof theorem

#### Authors and affiliations

• Paul-Hermann Zieschang
• 1
1. 1.Department of MathematicsUniversity of Texas at BrownsvilleBrownsvilleUSA