Bose–Mesner algebra

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.

Let X be a set of v elements. Consider a partition of the 2-element subsets of X into n non-empty subsets, R₁, ..., R_n 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 D_i defined by:

Let ${\displaystyle {\mathcal {A}}}$ be the vector space consisting of all matrices ${\displaystyle \sideset {}{_{i=0}^{n}}\sum a_{i}D_{i}}$, with ${\displaystyle a_{i}}$ complex.

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