In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative algebra.
Among these rules are: Bose–Mesner algebras have applications in physics to spin models, and in statistics to the design of experiments.
They are named for R. C. Bose and Dale Marsh Mesner.
[1] Let X be a set of v elements.
Consider a partition of the 2-element subsets of X into n non-empty subsets, R1, ..., Rn such that: This structure is enhanced by adding all pairs of repeated elements of X and collecting them in a subset R0.
This enhancement permits the parameters i, j, and k to take on the value of zero, and lets some of x,y or z be equal.
A set with such an enhanced partition is called an association scheme.
[2] One may view an association scheme as a partition of the edges of a complete graph (with vertex set X) into n classes, often thought of as color classes.
The association scheme can also be represented algebraically.
Consider the matrices Di defined by: Let
be the vector space consisting of all matrices
[3][4] The definition of an association scheme is equivalent to saying that the
are v × v (0,1)-matrices which satisfy The (x,y)-th entry of the left side of 4. is the number of two colored paths of length two joining x and y (using "colors" i and j) in the graph.
Note that the rows and columns of
are linearly independent, and the dimension of
This associative commutative algebra
is called the Bose–Mesner algebra of the association scheme.
are symmetric and commute with each other, they can be simultaneously diagonalized.
is semi-simple and has a unique basis of primitive idempotents
These are complex n × n matrices satisfying The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices
, and the basis consisting of the irreducible idempotent matrices
By definition, there exist well-defined complex numbers such that and The p-numbers
, play a prominent role in the theory.
[5] They satisfy well-defined orthogonality relations.
The p-numbers are the eigenvalues of the adjacency matrix
, satisfy the orthogonality conditions: Also In matrix notation, these are where
This implies that which proves Equation
The cases we are most interested in are those where the extended schemes are defined on the
The Kronecker power corresponds to the polynomial ring
Association schemes may be merged, but merging them leads to non-symmetric association schemes, whereas all usual codes are subgroups in symmetric Abelian schemes.