Matrix coefficient

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 groups 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 groups, in particular, reductive real and padic 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 systems are controlled by the properties of suitable matrix coefficients.
Contents
Definition
A matrix coefficient (or matrix element) of a linear representation ρ of a group G on a vector space V is a function f_{v,η} on the group, of the type
 f_{v,η}(g) = η(ρ(g).v)
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
 f_{v,w}(g) = <w,ρ(g).v>
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.
Applications
Finite groups
Matrix coefficients of irreducible representations of finite groups play a prominent role in representation theory of these groups, as developed by Burnside, Frobenius and Schur. They satisfy Schur orthogonality relations. The character of a representation ρ is a sum of the matrix coefficients f_{vi,ηi}, where {v_{i}} form a basis in the representation space of ρ, and {η_{i}} form the dual basis.
Finitedimensional Lie groups and special functions
Matrix coefficients of representations of Lie groups were first considered by Élie Cartan. Israel Gelfand realized that many classical special functions and orthogonal polynomials are expressible as the matrix coefficients of representation of Lie groups G.^{[1]} 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.^{[2]} 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.
Automorphic forms
A powerful approach to the theory of classical modular forms, initiated by Gelfand, Graev, and PiatetskiShapiro, views them as matrix coefficients of certain infinitedimensional unitary representations, automorphic representations of adelic groups. This approach was further developed by Langlands, for general reductive algebraic groups over global fields.
See also
 PeterWeyl theorem
 Spherical functions
Notes
 ^ Springer Online Reference Works
 ^ See the references for the complete treatment.
References
 Vilenkin, N. Ja. Special functions and the theory of group representations. Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22 American Mathematical Society, Providence, R. I. 1968
 Vilenkin, N. Ja., Klimyk, A. U. Representation of Lie groups and special functions. Recent advances. Translated from the Russian manuscript by V. A. Groza and A. A. Groza. Mathematics and its Applications, 316. Kluwer Academic Publishers Group, Dordrecht, 1995. xvi+497 pp. ISBN 0792332105
 Vilenkin, N. Ja., Klimyk, A. U. Representation of Lie groups and special functions. Vol. 3. Classical and quantum groups and special functions. Translated from the Russian by V. A. Groza and A. A. Groza. Mathematics and its Applications (Soviet Series), 75. Kluwer Academic Publishers Group, Dordrecht, 1992. xx+634 pp. ISBN 079231493X
 Vilenkin, N. Ja., Klimyk, A. U. Representation of Lie groups and special functions. Vol. 2. Class I representations, special functions, and integral transforms. Translated from the Russian by V. A. Groza and A. A. Groza. Mathematics and its Applications (Soviet Series), 74. Kluwer Academic Publishers Group, Dordrecht, 1993. xviii+607 pp. ISBN 0792314921
 Vilenkin, N. Ja., Klimyk, A. U. Representation of Lie groups and special functions. Vol. 1. Simplest Lie groups, special functions and integral transforms. Translated from the Russian by V. A. Groza and A. A. Groza. Mathematics and its Applications (Soviet Series), 72. Kluwer Academic Publishers Group, Dordrecht, 1991. xxiv+608 pp. ISBN 0792314662
Categories: Representation theory of groups
Wikimedia Foundation. 2010.
Look at other dictionaries:
Matrix mechanics — Quantum mechanics Uncertainty principle … Wikipedia
Matrix multiplication — In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n by m matrix and B is an m by p matrix, the result AB of their multiplication is an n by p matrix defined only if… … Wikipedia
Coefficient matrix — In linear algebra, the coefficient matrix refers to a matrix consisting of the coefficients of the variables in a set of linear equations. Example In general, a system with m linear equations and n unknowns can be written as … Wikipedia
Matrix exponential — In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.… … Wikipedia
Coefficient — For other uses of this word, see coefficient (disambiguation). In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the… … Wikipedia
Matrix differential equation — A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. A matrix differential equation is one containing more… … Wikipedia
Matrixfree methods — In computational mathematics, a matrix free method is an algorithm for solving a linear system of equations or an eigenvalue problem that does not store the coefficient matrix explicitly, but accesses the matrix by evaluating matrix vector… … Wikipedia
Multitraitmultimethod matrix — The multitrait multimethod (MTMM) matrix is an approach to examining Construct Validity developed by Campbell and Fiske(1959)[1]. There are six major considerations when examining a construct s validity through the MTMM matrix, which are as… … Wikipedia
Ceramic matrix composite — Fracture surface of a fiber reinforced ceramic composed of SiC fibers and SiC matrix. The fiber pull out mechanism shown is the key to CMC properties … Wikipedia
Transfermatrix method (optics) — The transfer matrix method is a method used in optics and acoustics to analyze the propagation of electromagnetic or acoustic waves through a stratified (layered) medium. [Born, M.; Wolf, E., Principles of optics: electromagnetic theory of… … Wikipedia