Bose–Mesner algebra - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Bose–Mesner algebra is a set of matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, together with set of rules (algebra) for combining (forming the products of) those matrices, such that certain conditions apply. Among these rules are:

the result of a product is also within the set of matrices
there is an identity matrix in the set
such that taking products is commutative
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

.

Bose–Mesner algebras have applications in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

to spin model

Spin model

A spin model is a mathematical model used in physics primarily to explain magnetism. Spin models may either be classical or quantum mechanical in nature. Spin models have been studied in quantum field theory as examples of integrable models. Spin models are also used in quantum information theory...

s, and in statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

to the design of experiments

Design of experiments

In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...

. They are named for R. C. Bose and Dale Marsh Mesner.

Background

Consider the vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

consisting of all matrices

Matrix (mathematics)

, with

real [1] [2] [3].

The definition of an association scheme

Association scheme

The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics,...

is equivalent to saying that the

are

-matrices

Matrix (mathematics)

which satisfy

I. is symmetric,
II. ,
III. ,
IV. .

The

-th entry of the left side of

is the number of paths in the graph. Note that the rows and columns of

contain

's:

From

, these matrices

Matrix (mathematics)

are symmetric. From

are linearly independent

Linear independence

In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...

, and the dimension of

. From

is closed under multiplication, and multiplication is always associative. This associative

Associative algebra

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

is called the Bose–Mesner algebra of the association scheme

Association scheme

. Since the matrices

Matrix (mathematics)

are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix

Matrix (mathematics)

such that to each

there is a diagonal matrix

Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

with

. This means that

is semi-simple and has a unique basis of primitive idempotents

. These are real

matrices

Matrix (mathematics)

satisfying

The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices

Adjacency matrix

In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...

, and the basis consisting of the irreducible idempotent matrices

Idempotent matrix

In algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix M is idempotent if and only if MM = M...

. By definition, there exist well-defined complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s such that

The p-numbers

, and the q-numbers

, play a prominent role in the theory [1]. They satisfy well-defined orthogonality relations. The p-numbers are the eigenvalues of the adjacency matrix

Adjacency matrix

In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...

Theorem

The eigenvalues of

and

, satisfy the orthogonality conditions:

Also

In matrix

Matrix (mathematics)

notation, these are

where

Proof of theorem

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 scheme

Association scheme

s and extensions

Kronecker's theorem

In mathematics, Kronecker's theorem is either of two theorems named after Leopold Kronecker.- The existence of extension fields :This is a theorem stating that a polynomial in a field, p ∈ F[x], has a root in an extension field E \supset F.For example, a polynomial in the reals such...

of finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s. 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

Association scheme

is defined. A first association scheme

Association scheme

defined on

is called the

-th Kronecker power

Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...

. Next the extension is defined on the same set

by gathering classes of

. The Kronecker power

Kronecker product

corresponds to the polynomial ring

Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

first defined on a field

Finite 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

Hamming scheme

The Hamming scheme, named after Richard Hamming, is also known as the hyper-cubic association scheme, and it is the most important example for coding theory...

.

Association scheme

Association scheme

s may be merged, but merging them leads to non-symmetric association scheme

Association scheme

s, whereas all usual code

Code

A code is a rule for converting a piece of information into another form or representation , not necessarily of the same type....

s are subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s in symmetric Abelian schemes

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...

[1] [2] [3].

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