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