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:
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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
The definition of an association scheme is equivalent to saying that the
-
D i -
∑ i = 0 n D i = J (the all-ones matrix), -
D 0 = I , -
D i D j = ∑ k = 0 n p i j k D k = D j D i , i , j = 0 , … , n .
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
From 1., these matrices are symmetric. From 2.,
The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices
and
The p-numbers
Theorem
The eigenvalues of
Also
In matrix notation, these are
where
Proof of theorem
The eigenvalues of
which proves Equation
which gives Equations
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
Association schemes may be merged, but merging them leads to non-symmetric association schemes, whereas all usual codes are subgroups in symmetric Abelian schemes.