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:
- the result of a product is also within the set of matrices,
- there is an identity matrix in the set, and
- taking products is commutative.
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:
- given an , the number of such that depends only on i (and not on x). This number will be denoted by vi, and
- given with , the number of such that and depends only on i,j and k (and not on x and y). This number will be denoted by .
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:
- is symmetric,
- (the all-ones matrix),
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 contain 1s:
From 1., these matrices are symmetric. From 2., are linearly independent, and the dimension of is . From 4., is closed under multiplication, and multiplication is always associative. This associative commutative algebra is called the Bose–Mesner algebra of the association scheme. Since the matrices in are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix such that to each there is a diagonal matrix with . This means that 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
The eigenvalues of and , satisfy the orthogonality conditions:
In matrix notation, these are
Proof of theoremEdit
The eigenvalues of are with multiplicities . This implies that
which proves Equation and Equation ,
which gives Equations , and .
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 -th Cartesian power of a set on which a basic association scheme is defined. A first association scheme defined on is called the -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 , while the extension scheme corresponds to the extension field obtained as a quotient. An example of such an extended scheme is the Hamming scheme.
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (September 2010) (Learn how and when to remove this template message)
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