Matrix coefficient
Encyclopedia
In mathematics
, a matrix coefficient (or matrix element) is a function on a
group
of a special form, which depends on a linear representation of the group and additional data. For the case of a finite group, matrix coefficients express the action of the elements of the group in the specified representation via the entries of the corresponding matrices
.
Matrix coefficients of representations of Lie group
s turned out to be intimately related with the theory of special functions
, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play key role in the classification of irreducible representations of locally compact group
s, in particular, reductive real and p-adic groups. The formalism of matrix coefficients leads to a vast generalization of the notion of a modular form
. In a different direction, mixing
properties of certain dynamical system
s are controlled by the properties of suitable matrix coefficients.
V is a function fv,η on the group, of the type
where v is a vector in V, η is a continuous linear functional
on V, and g is an element of G. This function takes scalar values on G. If V is a Hilbert space
, then by the Riesz representation theorem
, all matrix coefficients have the form
for some vectors v and w in V.
For V of finite dimension, and v and w taken from a standard basis
, this is actually the function given by the matrix
entry in a fixed place.
, Frobenius and Schur
. They satisfy Schur orthogonality relations
. The character
of a representation ρ is a sum of the matrix coefficients fvi,ηi, where {vi} form a basis in the representation space of ρ, and {ηi} form the dual basis.
.
Israel Gelfand
realized that many classical special functions
and orthogonal polynomials
are expressible as the matrix coefficients of representation of Lie groups G. This description provides a uniform framework for proving many hitherto disparate properties of special functions, such as addition formulas, certain recurrence relations, orthogonality relations, integral representations, and eigenvalue properties with respect to differential operators. Special functions of mathematical physics, such as the trigonometric functions, the hypergeometric function and its generalizations, Legendre and Jacobi
orthogonal polynomials and Bessel functions all arise as matrix coefficients of representations of Lie groups. Theta functions and real analytic Eisenstein series
, important in algebraic geometry
and number theory
, also admit such realizations.
s, initiated by Gelfand, Graev, and Piatetski-Shapiro, views them as matrix coefficients of certain infinite-dimensional unitary representations, automorphic representations of adelic groups. This approach was further developed
by Langlands
, for general reductive algebraic groups over global field
s.
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 matrix coefficient (or matrix element) is a function on a
group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of a special form, which depends on a linear representation of the group and additional data. For the case of a finite group, matrix coefficients express the action of the elements of the group in the specified representation via the entries of the corresponding 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...
.
Matrix coefficients of representations of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s turned out to be intimately related with the theory of special functions
Special functions
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....
, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play key role in the classification of irreducible representations of locally compact group
Locally compact group
In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
s, in particular, reductive real and p-adic groups. The formalism of matrix coefficients leads to a vast generalization of the notion of a modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
. In a different direction, mixing
Mixing (mathematics)
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc....
properties of certain dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s are controlled by the properties of suitable matrix coefficients.
Definition
A matrix coefficient (or matrix element) of a linear representation ρ of a group G on a vector spaceVector 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...
V is a function fv,η on the group, of the type
- fv,η(g) = η(ρ(g).v)
where v is a vector in V, η is a continuous linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
on V, and g is an element of G. This function takes scalar values on G. If V is a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, then by the Riesz representation theorem
Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...
, all matrix coefficients have the form
- fv,w(g) =
for some vectors v and w in V.
For V of finite dimension, and v and w taken from a standard basis
Standard basis
In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...
, this is actually the function given by the matrix
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...
entry in a fixed place.
Finite groups
Matrix coefficients of irreducible representations of finite groups play a prominent role in representation theory of these groups, as developed by BurnsideWilliam Burnside
William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....
, Frobenius and Schur
Issai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
. They satisfy Schur orthogonality relations
Schur orthogonality relations
In mathematics, the Schur orthogonality relations express a central fact about representations of finite groups.They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as therotation group SO....
. The character
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
of a representation ρ is a sum of the matrix coefficients fvi,ηi, where {vi} form a basis in the representation space of ρ, and {ηi} form the dual basis.
Finite-dimensional Lie groups and special functions
Matrix coefficients of representations of Lie groups were first considered by Élie CartanÉlie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
.
Israel Gelfand
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...
realized that many classical special functions
Special functions
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....
and orthogonal polynomials
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
are expressible as the matrix coefficients of representation of Lie groups G. This description provides a uniform framework for proving many hitherto disparate properties of special functions, such as addition formulas, certain recurrence relations, orthogonality relations, integral representations, and eigenvalue properties with respect to differential operators. Special functions of mathematical physics, such as the trigonometric functions, the hypergeometric function and its generalizations, Legendre and Jacobi
Jacobi polynomials
In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...
orthogonal polynomials and Bessel functions all arise as matrix coefficients of representations of Lie groups. Theta functions and real analytic Eisenstein series
Real analytic Eisenstein series
In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL and in analytic number theory...
, important in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, also admit such realizations.
Automorphic forms
A powerful approach to the theory of classical modular formModular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s, initiated by Gelfand, Graev, and Piatetski-Shapiro, views them as matrix coefficients of certain infinite-dimensional unitary representations, automorphic representations of adelic groups. This approach was further developed
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
by Langlands
Robert Langlands
Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...
, for general reductive algebraic groups over global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
s.