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.

Definition

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. 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. In this representation, there is a loop at each vertex and all the loops receive the same 0th color. The association scheme can also be represented algebraically. Consider the matrices Di defined by: Let \mathcal{A} be the vector space consisting of all matrices, with a_{i} complex. The definition of an association scheme is equivalent to saying that the D_{i} 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 D_i contain v_i 1s: From 1., these matrices are symmetric. From 2., are linearly independent, and the dimension of \mathcal{A} is n+1. From 4., \mathcal{A} is closed under multiplication, and multiplication is always associative. This associative commutative algebra \mathcal{A} is called the Bose–Mesner algebra of the association scheme. Since the matrices in \mathcal{A} are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix S such that to each there is a diagonal matrix \Lambda_{A} with. This means that \mathcal{A} 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 D_i, and the basis consisting of the irreducible idempotent matrices E_k. By definition, there exist well-defined complex numbers such that and The p-numbers p_i (k), and the q-numbers q_k(i), play a prominent role in the theory. They satisfy well-defined orthogonality relations. The p-numbers are the eigenvalues of the adjacency matrix D_i.

Theorem

The eigenvalues of p_i(k) and q_k(i), satisfy the orthogonality conditions: Also In matrix notation, these are where

Proof of theorem

The eigenvalues of D_i D_\ell are with multiplicities \mu_k. This implies that which proves Equation and Equation , which gives Equations (9), (10) and (12).\Box There is an analogy between extensions of association schemes and extensions of finite fields. The cases we are most interested in are those where the extended schemes are defined on the n-th Cartesian power of a set \mathcal{F} on which a basic association scheme is defined. A first association scheme defined on is called the n-th Kronecker power of. Next the extension is defined on the same set by gathering classes of. The Kronecker power corresponds to the polynomial ring first defined on a field \mathbb{F}, while the extension scheme corresponds to the extension field obtained as a quotient. An example of such an extended scheme is the Hamming scheme. Association schemes may be merged, but merging them leads to non-symmetric association schemes, whereas all usual codes are subgroups in symmetric Abelian schemes.