Association scheme

History

Origin of the concept

The concept of assication scheme is inherent in the 1939 paper by Bose and Nair.

Origin of the term

The term association scheme was introduced in the 1952 paper by Bose and Shimamoto.

Definition

An association scheme on a set is a partition of the set of unordered pairs into subsets $A_{i}$ such that given any $i,j,h$, and any $(x,y)\in A_{h}$, the number of $z$ such that $(x,z)\in A_{i}$ and $(z,y)\in A_{j}$ depends only on $i,j,h$ and not on $x,y$.

Examples

Coset space of a group

Given a group and a subgroup, we can define the double cosets of the subgroup as relative positions of pairs of left cosets. Thus the set of pairs of left cosets can be partitioned into subsets based on relative position. The fact that this partition forms an association scheme is crucial in defining the Hecke algebra for the subgroup.

References

Papers

• Partially balanced incomplete block designs by R. C. Bose and K. R. Nair, Sankhya 4, 337-372, 1939
• Classification and analysis of partially balanced incomplete block designs with two associate classes by R. C. Bose and T. Shimamoto, J. Amer. Statistical Association, 47, 151-184, 1952

Texts

• Finite Geometries by Peter Dembowski, Appendix 7.1 Association schemes and partial designs