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{{Short description|Functions on special groups related to their matrix representations}}
In [[mathematics]], a '''matrix coefficient''' (or '''matrix element''') is a function on a [[group (mathematics)|group]] of a special form, which depends on a [[linear representation]] of the group and additional data. Precisely, it is a function on a [[compact group|compact topological group]] ''G'' obtained by [[Function composition|composing]] a representation of ''G'' on a [[vector space]] ''V'' with a [[linear map]] from the [[endomorphism]]s of ''V'' into ''V''{{'s}} underlying [[field (mathematics)|field]]. It is also called a '''representative function'''.{{sfn|Bröcker |tom Dieck|1985}} They arise naturally from finite-dimensional representations of ''G'' as the [[matrix (mathematics)|matrix]]-entry functions of the corresponding matrix representations. The [[Peter–Weyl theorem]] says that the matrix coefficients on ''G'' are dense in the [[Hilbert space]] of square-integrable functions on ''G''.

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 a key role in the classification of [[irreducible representations]] of [[locally compact group]]s, in particular, reductive real and [[p-adic|''p''-adic]] groups. The formalism of matrix coefficients leads to a generalization of the notion of a [[modular form]]. In a different direction, [[mixing (mathematics)|mixing]] properties of certain [[dynamical system]]s are controlled by the properties of suitable matrix coefficients.

== Definition ==
A '''matrix coefficient''' (or '''matrix element''') of a linear representation {{math|ρ}} of a group {{math|G}} on a [[vector space]] {{math|V}} is a function {{math|fv,η}} on the group, of the type

:<math>f_{v,\eta}(g) = \eta(\rho(g)v)</math>

where {{math|v}} is a vector in {{math|V}}, {{math|η}} is a continuous [[linear functional]] on {{math|V}}, and {{math|g}} is an element of {{math|G}}. This function takes scalar values on {{math|G}}. If {{math|V}} is a [[Hilbert space]], then by the [[Riesz representation theorem]], all matrix coefficients have the form

:<math>f_{v,w}(g) = \langle \rho(g)v, w \rangle</math>

for some vectors {{math|v}} and {{math|w}} in {{math|V}}.

For {{math|V}} of finite dimension, and {{math|v}} and {{math|w}} taken from a [[standard basis]], this is actually the function given by the [[matrix (mathematics)|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 [[William Burnside|Burnside]], [[Georg Frobenius|Frobenius]] and [[Issai Schur|Schur]]. They satisfy [[Schur orthogonality relations]]. The [[character theory|character]] of a representation ρ is a sum of the matrix coefficients ''f''''v''i,ηi, where {''v''i} 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 Cartan]].
[[Israel Gelfand]] realized that many classical [[special functions]] and [[orthogonal polynomials]] are expressible as the matrix coefficients of representation of Lie groups ''G''.<ref>{{SpringerEOM|title=Special functions}}</ref>{{Citation needed| reason= What's given is not a citation|date=October 2020}} 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.<ref>See the references for the complete treatment.</ref> Special functions of mathematical physics, such as the [[trigonometric functions]], the [[hypergeometric function]] and its generalizations, [[Legendre polynomials|Legendre]] and [[Jacobi polynomials|Jacobi]] orthogonal polynomials and [[Bessel functions]] all arise as matrix coefficients of representations of Lie groups. [[Theta function]]s 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 form]]s, initiated by Gelfand, [[Mark Iosifovich Graev|Graev]], and [[Piatetski-Shapiro]], views them as matrix coefficients of certain infinite-dimensional unitary representations, [[automorphic representation]]s of [[adelic group]]s. This approach was [[Langlands program|further developed]] by [[Robert Langlands|Langlands]], for general [[reductive algebraic group]]s over [[global field]]s.

== See also ==
*[[Peter–Weyl theorem]]
*[[Spherical functions]]
*[[Discrete series representation]]

==Notes==
<references/>

== References ==
* {{cite book|first1=Theodor |last1=Bröcker|first2= Tammo |last2=tom Dieck|title=Representations of compact Lie groups|series= Graduate Texts in Mathematics|volume=98|publisher= [[Springer-Verlag]]|location= Berlin|year =1985|mr=0781344 |isbn=0-387-13678-9}}
* {{cite book | last1=Hochschild | first1=G. |author1-link=Gerhard Hochschild | title=The Structure of Lie Groups | publisher=Holden-Day| location=San Francisco, London, Amsterdam | mr=0207883 | year=1965}}
*[[Naum Vilenkin|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|0-7923-3210-5}}
* 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|0-7923-1493-X}}
* 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|0-7923-1492-1}}
* 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|0-7923-1466-2}}
* {{cite book|last=Želobenko|first= D. P.|title=Compact Lie groups and their representations|series=Translations of Mathematical Monographs|volume=40|publisher=[[American Mathematical Society]]|year= 1973}}

[[Category:Representation theory of groups]]

Matrix coefficient - Revision history

imported>Tamzin: change '' 's → ''{{'s}} (via WP:JWB)