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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Class function</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with a <a href="Class_(set_theory)#Classes_in_formal_set_theories" title="Class (set theory)">class function</a> in set theory.</div>
<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, especially in the fields of <a href="Group_theory" title="Group theory">group theory</a> and <a href="Group_representation" title="Group representation">representation theory of groups</a>, a <b>class function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> on a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> that is constant on the <a href="Conjugacy_class" title="Conjugacy class">conjugacy classes</a> of <i>G</i>. In other words, it is invariant under the <a href="Conjugation_map" class="mw-redirect" title="Conjugation map">conjugation map</a> on <i>G</i>. Such functions play a basic role in <a href="Representation_theory" title="Representation theory">representation theory</a>.
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<div class="mw-heading mw-heading2"><h2 id="Characters">Characters</h2></div>
<p>The <a href="Character_(group_theory)" class="mw-redirect" title="Character (group theory)">character</a> of a <a href="Linear_representation" class="mw-redirect" title="Linear representation">linear representation</a> of <i>G</i> over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <i>K</i> is always a class function with values in <i>K</i>. The class functions form the <a href="Center_(ring_theory)" title="Center (ring theory)">center</a> of the <a href="Group_ring" title="Group ring">group ring</a> <i>K</i>[<i>G</i>]. Here a class function <i>f</i> is identified with the element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{g\in G}f(g)g}">
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<mo>∑<!-- ∑ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \sum _{g\in G}f(g)g}</annotation>
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</math></span><img src="./4029c250e0edb5e54d71bcb402344198fb88d786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:9.062ex; height:5.843ex;" alt="{\displaystyle \sum _{g\in G}f(g)g}" loading="lazy"></span>.
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<div class="mw-heading mw-heading2"><h2 id="Inner_products">Inner products</h2></div>
<p>The set of class functions of a group <span class="texhtml mvar" style="font-style:italic;">G</span> with values in a field <span class="texhtml mvar" style="font-style:italic;">K</span> form a <span class="texhtml mvar" style="font-style:italic;">K</span>-<a href="Vector_space" title="Vector space">vector space</a>. If <span class="texhtml mvar" style="font-style:italic;">G</span> is finite and the <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> of the field does not divide the order of <span class="texhtml mvar" style="font-style:italic;">G</span>, then there is an <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> defined on this space defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi ,\psi \rangle ={\frac {1}{|G|}}\sum _{g\in G}\phi (g){\overline {\psi (g)}},}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>ϕ<!-- ϕ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \langle \phi ,\psi \rangle ={\frac {1}{|G|}}\sum _{g\in G}\phi (g){\overline {\psi (g)}},}</annotation>
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</math></span><img src="./99594c138a6d69b20dccf7c6886296c0d29cee8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.437ex; height:6.676ex;" alt="{\displaystyle \langle \phi ,\psi \rangle ={\frac {1}{|G|}}\sum _{g\in G}\phi (g){\overline {\psi (g)}},}" loading="lazy"></span> where <span class="texhtml">|<i>G</i>|</span> denotes the order of <span class="texhtml mvar" style="font-style:italic;">G</span> and the overbar denotes conjugation in the field <span class="texhtml mvar" style="font-style:italic;">K</span>. The set of <a href="Irreducible_character" class="mw-redirect" title="Irreducible character">irreducible characters</a> of <span class="texhtml mvar" style="font-style:italic;">G</span> forms an <a href="Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a>. Further, if <span class="texhtml mvar" style="font-style:italic;">K</span> is a <a href="Splitting_field" title="Splitting field">splitting field</a> for <span class="texhtml mvar" style="font-style:italic;">G</span>—for instance, if <span class="texhtml mvar" style="font-style:italic;">K</span> is <a href="Algebraically_closed" class="mw-redirect" title="Algebraically closed">algebraically closed</a>, then the irreducible characters form an <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>.
</p><p>When <span class="texhtml mvar" style="font-style:italic;">G</span> is a <a href="Compact_group" title="Compact group">compact group</a> and <span class="texhtml"><i>K</i> = <b>C</b></span> is the field of <a href="Complex_number" title="Complex number">complex numbers</a>, the <a href="Haar_measure" title="Haar measure">Haar measure</a> can be applied to replace the finite sum above with an integral: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi ,\psi \rangle =\int _{G}\phi (t){\overline {\psi (t)}}\,dt.}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \langle \phi ,\psi \rangle =\int _{G}\phi (t){\overline {\psi (t)}}\,dt.}</annotation>
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</math></span><img src="./7f2b1da4b0aea299e5c6c5728c3971d282396db5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.445ex; height:5.676ex;" alt="{\displaystyle \langle \phi ,\psi \rangle =\int _{G}\phi (t){\overline {\psi (t)}}\,dt.}" loading="lazy"></span>
</p><p>When <span class="texhtml mvar" style="font-style:italic;">K</span> is the real numbers or the complex numbers, the inner product is a <a href="Degenerate_form" class="mw-redirect" title="Degenerate form">non-degenerate</a> <a href="Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian</a> <a href="Bilinear_form" title="Bilinear form">bilinear form</a>.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Brauer's_theorem_on_induced_characters" title="Brauer's theorem on induced characters">Brauer's theorem on induced characters</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a>, <i>Linear representations of finite groups</i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a> <b>42</b>, Springer-Verlag, Berlin, 1977.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-02-27" href="https://en.wikipedia.org/wiki/?title=Class_function&oldid=1277893305">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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