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. 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.
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:
An association scheme is "commutative" if formula_13 for all formula_8, formula_9 and formula_10. Most authors assume this property.
A "symmetric" association scheme is one in which each relation formula_17 is a symmetric relation. That is:
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 formula_18. 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 formula_8. 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 formula_20 which are both "i" th associates of formula_11 and "j" th associates of formula_12 is a constant formula_7.
Graph interpretation and adjacency matrices.
A symmetric association scheme can be visualized as a complete graph with labeled edges. The graph has formula_24 vertices, one for each point of formula_25, and the edge joining vertices formula_11 and formula_12 is labeled formula_8 if formula_11 and formula_12 are formula_8 th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled formula_10 having the other edges labeled formula_8 and formula_9 is a constant formula_7, depending on formula_36 but not on the choice of the base. In particular, each vertex is incident with exactly formula_37 edges labeled formula_8; formula_39 is the valency of the relation formula_40. There are also loops labeled formula_41 at each vertex formula_11, corresponding to formula_43.
The relations are described by their adjacency matrices. formula_44 is the adjacency matrix of formula_40 for formula_46 and is a "v" × "v" matrix with rows and columns labeled by the points of formula_25.
The definition of a symmetric association scheme is equivalent to saying that the formula_44 are "v" × "v" (0,1)-matrices which satisfy
The ("x", "y")-th entry of the left side of (IV) is the number of paths of length two between "x" and "y" with labels i and j in the graph. Note that the rows and columns of formula_44 contain formula_39 formula_56's:
formula_57
History.
The term "association scheme" is due to but the concept is already inherent in . These authors were studying what statisticians have called "partially balanced incomplete block designs" (PBIBDs). The subject became an object of algebraic interest with the publication of and the introduction of the Bose–Mesner algebra. The most important contribution to the theory was the thesis of P. Delsarte who recognized and fully used the connections with coding theory and design theory. Generalizations have been studied by D. G. Higman (coherent configurations) and B. Weisfeiler (distance regular graphs).
The Bose–Mesner algebra.
The adjacency matrices formula_68 of the graphs formula_69 generate a commutative and associative algebra formula_70 (over the real or complex numbers) both for the matrix product and the pointwise product. This associative, commutative algebra is called the Bose–Mesner algebra of the association scheme.
Since the matrices in formula_70 are symmetric and commute with each other, they can be diagonalized simultaneously. Therefore, formula_70 is semi-simple and has a unique basis of primitive idempotents formula_73.
There is another algebra of formula_74 matrices which is isomorphic to formula_70, and is often easier to work with.
Coding theory.
The Hamming scheme and the Johnson scheme are of major significance in classical coding theory.
In coding theory, association scheme theory is mainly concerned with the distance of a code. The linear programming method produces upper bounds for the size of a code with given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain polynomial properties; this leads one into the realm of orthogonal polynomials. In particular, some universal bounds are derived for codes and designs in polynomial-type association schemes.
In classical coding theory, dealing with codes in a Hamming scheme, the MacWilliams transform involves a family of orthogonal polynomials known as the Krawtchouk polynomials. These polynomials give the eigenvalues of the distance relation matrices of the Hamming scheme.