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. Definition. Let "X" be a set of "v" elements. Consider a partition  of the 2-element subsets of "X" into "n" non-empty subsets, "R", ..., "R" such that: This structure is enhanced by adding all pairs of repeated elements of "X" and collecting them in a subset "R". 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" defined by: Let formula_11 be the vector space consisting of all matrices formula_12, with formula_13 complex. The definition of an association scheme is equivalent to saying that the formula_14 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 formula_15 contain formula_20 1s: From 1., these matrices are symmetric. From 2., formula_22 are linearly independent, and the dimension of formula_11 is formula_24. From 4., formula_11 is closed under multiplication, and multiplication is always associative. This associative commutative algebra formula_11 is called the Bose–Mesner algebra of the association scheme. Since the matrices in formula_11 are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix formula_28 such that to each formula_29 there is a diagonal matrix formula_30 with formula_31. This means that formula_11 is  semi-simple and has a unique basis of primitive idempotents formula_33. These are complex n × n matrices satisfying The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices formula_15, and the basis consisting of the irreducible idempotent matrices formula_38. By definition, there exist well-defined complex numbers such that and The p-numbers formula_41, and the q-numbers formula_42, play a prominent role in the theory. They satisfy well-defined orthogonality relations. The p-numbers are the eigenvalues of the adjacency matrix formula_15. Theorem. The eigenvalues of formula_44 and formula_42, satisfy the orthogonality conditions : Also In matrix notation, these are where formula_51 Proof of theorem. The eigenvalues of formula_52 are formula_53 with multiplicities formula_54. This implies that which proves Equation formula_56 and Equation formula_57, which gives Equations formula_59, formula_60 and formula_61.formula_62 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 formula_63-th Cartesian power formula_64 of a set formula_65 on which a basic association scheme formula_66 is defined. A first association scheme defined on formula_64 is called the formula_63-th Kronecker power  formula_69 of formula_66. Next the extension is defined on the same set formula_64 by gathering classes of formula_69. The Kronecker power corresponds to the polynomial ring formula_73 first defined on a field formula_74, 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.