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<title>Matrix coefficient</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Matrix coefficient</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>matrix coefficient</b> (or <b>matrix element</b>) is a function on a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> of a special form, which depends on a <a href="Linear_representation" class="mw-redirect" title="Linear representation">linear representation</a> of the group and additional data. Precisely, it is a function on a <a href="Compact_group" title="Compact group">compact topological group</a> <i>G</i> obtained by <a href="Function_composition" title="Function composition">composing</a> a representation of <i>G</i> on a <a href="Vector_space" title="Vector space">vector space</a> <i>V</i> with a <a href="Linear_map" title="Linear map">linear map</a> from the <a href="Endomorphism" title="Endomorphism">endomorphisms</a> of <i>V</i> into <i>V</i><span class="nowrap" style="padding-left:0.1em;">'s</span> underlying <a href="Field_(mathematics)" title="Field (mathematics)">field</a>. It is also called a <b>representative function</b>.<sup id="cite_ref-FOOTNOTEBröckertom_Dieck1985_1-0" class="reference"><a href="#cite_note-FOOTNOTEBröckertom_Dieck1985-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> They arise naturally from finite-dimensional representations of <i>G</i> as the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>-entry functions of the corresponding matrix representations. The <a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a> says that the matrix coefficients on <i>G</i> are dense in the <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> of square-integrable functions on <i>G</i>.

</p><p>Matrix coefficients of representations of <a href="Lie_group" title="Lie group">Lie groups</a> turned out to be intimately related with the theory of <a href="Special_functions" title="Special functions">special functions</a>, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification of <a href="Irreducible_representations" class="mw-redirect" title="Irreducible representations">irreducible representations</a> of <a href="Locally_compact_group" title="Locally compact group">locally compact groups</a>, in particular, reductive real and <a href="P-adic" class="mw-redirect" title="P-adic"><i>p</i>-adic</a> groups. The formalism of matrix coefficients leads to a generalization of the notion of a <a href="Modular_form" title="Modular form">modular form</a>. In a different direction, <a href="Mixing_(mathematics)" title="Mixing (mathematics)">mixing</a> properties of certain <a href="Dynamical_system" title="Dynamical system">dynamical systems</a> are controlled by the properties of suitable matrix coefficients.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>A <b>matrix coefficient</b> (or <b>matrix element</b>) of a linear representation <span class="texhtml">ρ</span> of a group <span class="texhtml">G</span> on a <a href="Vector_space" title="Vector space">vector space</a> <span class="texhtml">V</span> is a function <span class="texhtml">f<sub>v,η</sub></span> on the group, of the type

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<annotation encoding="application/x-tex">{\displaystyle f_{v,\eta }(g)=\eta (\rho (g)v)}</annotation>

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</math></span><img src="./c143686c15aa32d65b0e2661f1eae3680d00f04c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.71ex; height:3.009ex;" alt="{\displaystyle f_{v,\eta }(g)=\eta (\rho (g)v)}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml">v</span> is a vector in <span class="texhtml">V</span>, <span class="texhtml">η</span> is a continuous <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on <span class="texhtml">V</span>, and <span class="texhtml">g</span> is an element of <span class="texhtml">G</span>. This function takes scalar values on <span class="texhtml">G</span>. If <span class="texhtml">V</span> is a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>, then by the <a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a>, all matrix coefficients have the form

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<annotation encoding="application/x-tex">{\displaystyle f_{v,w}(g)=\langle \rho (g)v,w\rangle }</annotation>

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</math></span><img src="./26b83dcc05d338fed91d5f3aa9042d815b2d693f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.589ex; height:3.009ex;" alt="{\displaystyle f_{v,w}(g)=\langle \rho (g)v,w\rangle }" loading="lazy"></span></dd></dl>

<p>for some vectors <span class="texhtml">v</span> and <span class="texhtml">w</span> in <span class="texhtml">V</span>.

</p><p>For <span class="texhtml">V</span> of finite dimension, and <span class="texhtml">v</span> and <span class="texhtml">w</span> taken from a <a href="Standard_basis" title="Standard basis">standard basis</a>, this is actually the function given by the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> entry in a fixed place.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Finite_groups">Finite groups</h3></div>

<p>Matrix coefficients of irreducible representations of finite groups play a prominent role in representation theory of these groups, as developed by <a href="William_Burnside" title="William Burnside">Burnside</a>, <a href="Georg_Frobenius" class="mw-redirect" title="Georg Frobenius">Frobenius</a> and <a href="Issai_Schur" title="Issai Schur">Schur</a>. They satisfy <a href="Schur_orthogonality_relations" title="Schur orthogonality relations">Schur orthogonality relations</a>. The <a href="Character_theory" title="Character theory">character</a> of a representation ρ is a sum of the matrix coefficients <i>f</i><sub><i>v</i><sub>i</sub>,η<sub>i</sub></sub>, where {<i>v</i><sub>i</sub>} form a basis in the representation space of ρ, and {η<sub>i</sub>} form the <a href="Dual_basis" title="Dual basis">dual basis</a>.

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<div class="mw-heading mw-heading3"><h3 id="Finite-dimensional_Lie_groups_and_special_functions">Finite-dimensional Lie groups and special functions</h3></div>

<p>Matrix coefficients of representations of Lie groups were first considered by <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>.

<a href="Israel_Gelfand" title="Israel Gelfand">Israel Gelfand</a> realized that many classical <a href="Special_functions" title="Special functions">special functions</a> and <a href="Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a> are expressible as the matrix coefficients of representation of Lie groups <i>G</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 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 <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> properties with respect to differential operators.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Special functions of mathematical physics, such as the <a href="Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, the <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a> and its generalizations, <a href="Legendre_polynomials" title="Legendre polynomials">Legendre</a> and <a href="Jacobi_polynomials" title="Jacobi polynomials">Jacobi</a> orthogonal polynomials and <a href="Bessel_functions" class="mw-redirect" title="Bessel functions">Bessel functions</a> all arise as matrix coefficients of representations of Lie groups. <a href="Theta_function" title="Theta function">Theta functions</a> and <a href="Real_analytic_Eisenstein_series" title="Real analytic Eisenstein series">real analytic Eisenstein series</a>, important in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and <a href="Number_theory" title="Number theory">number theory</a>, also admit such realizations.

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<div class="mw-heading mw-heading3"><h3 id="Automorphic_forms">Automorphic forms</h3></div>

<p>A powerful approach to the theory of classical <a href="Modular_form" title="Modular form">modular forms</a>, initiated by Gelfand, <a href="Mark_Iosifovich_Graev" title="Mark Iosifovich Graev">Graev</a>, and <a href="Piatetski-Shapiro" class="mw-redirect" title="Piatetski-Shapiro">Piatetski-Shapiro</a>, views them as matrix coefficients of certain infinite-dimensional unitary representations, <a href="Automorphic_representation" class="mw-redirect" title="Automorphic representation">automorphic representations</a> of <a href="Adelic_group" class="mw-redirect" title="Adelic group">adelic groups</a>. This approach was <a href="Langlands_program" title="Langlands program">further developed</a> by <a href="Robert_Langlands" title="Robert Langlands">Langlands</a>, for general <a href="Reductive_algebraic_group" class="mw-redirect" title="Reductive algebraic group">reductive algebraic groups</a> over <a href="Global_field" title="Global field">global fields</a>.

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<ul><li><a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a></li>

<li><a href="Spherical_functions" class="mw-redirect" title="Spherical functions">Spherical functions</a></li>

<li><a href="Discrete_series_representation" title="Discrete series representation">Discrete series representation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-FOOTNOTEBröckertom_Dieck1985-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBröckertom_Dieck1985_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBröckertom_Dieck1985">Bröcker & tom Dieck 1985</a>.</span>

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</style><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Special_functions">"Special functions"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">See the references for the complete treatment.</span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFBröckertom_Dieck1985" class="citation book cs1">Bröcker, Theodor; tom Dieck, Tammo (1985). <i>Representations of compact Lie groups</i>. Graduate Texts in Mathematics. Vol. 98. Berlin: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-13678-9</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0781344">0781344</a>.</cite></li>

<li><cite id="CITEREFHochschild1965" class="citation book cs1"><a href="Gerhard_Hochschild" title="Gerhard Hochschild">Hochschild, G.</a> (1965). <i>The Structure of Lie Groups</i>. San Francisco, London, Amsterdam: Holden-Day. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0207883">0207883</a>.</cite></li>

<li><a href="Naum_Vilenkin" class="mw-redirect" title="Naum Vilenkin">Vilenkin, N. Ja.</a> <i>Special functions and the theory of group representations</i>. Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22 American Mathematical Society, Providence, R. I. 1968</li>

<li>Vilenkin, N. Ja., Klimyk, A. U. <i>Representation of Lie groups and special functions. Recent advances</i>. 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. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-3210-5</bdi></li>

<li>Vilenkin, N. Ja., Klimyk, A. U. <i>Representation of Lie groups and special functions. Vol. 3. Classical and quantum groups and special functions</i>. 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. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-1493-X</bdi></li>

<li>Vilenkin, N. Ja., Klimyk, A. U. <i>Representation of Lie groups and special functions. Vol. 2. Class I representations, special functions, and integral transforms</i>. 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. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-1492-1</bdi></li>

<li>Vilenkin, N. Ja., Klimyk, A. U. <i>Representation of Lie groups and special functions. Vol. 1. Simplest Lie groups, special functions and integral transforms</i>. 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. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-1466-2</bdi></li>

<li><cite id="CITEREFŽelobenko1973" class="citation book cs1">Želobenko, D. P. (1973). <i>Compact Lie groups and their representations</i>. Translations of Mathematical Monographs. Vol. 40. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>.</cite></li></ul></div><div><div class="zim-footer">

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