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, ..., Rn 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) ∈ Ri, then (y,x) ∈ Ri . (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 .
Read more about this topic: Association Scheme
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