Association scheme

The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.

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:

An association scheme is commutative if ${\displaystyle p_{ij}^{k}=p_{ji}^{k}}$ for all ${\displaystyle i}$, ${\displaystyle j}$ and ${\displaystyle k}$. Most authors assume this property.

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