Association Scheme - Definition

Definition

An n-class association scheme consists of a set X together with a partition S of X × X into n + 1 binary relations, R₀, R₁, ..., R_n which satisfy:

• and is called the Identity relation.
• Defining, if R in S, then R* in S
• If, the number of such that and is a constant depending on, but not on the particular choice of and .

An association scheme is commutative if for all, and . Most authors assume this property.

A symmetric association scheme is one in which each relation is a symmetric relation. That is:

• if (x,y) ∈ R_i, then (y,x) ∈ R_i . (Or equivalently, R* = R.)

Every symmetric association scheme is commutative.

Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group.

Two points x and y are called i th associates if . The definition states that if x and y are i th associates so are y and x. Every pair of points are i th associates for exactly one . Each point is its own zeroth associate while distinct points are never zeroth associates. If x and y are k th associates then the number of points which are both i th associates of and j th associates of is a constant .