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<title>Orthogonalitätsrelationen</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Orthogonalitätsrelationen</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Im <a href="Teilgebiet_der_Mathematik" class="mw-redirect" title="Teilgebiet der Mathematik">mathematischen Teilgebiet</a> der <a href="Gruppentheorie" title="Gruppentheorie">Gruppentheorie</a> sind die Orthogonalitätsrelationen bestimmte Beziehungen zwischen <a href="Charakter_(Mathematik)" title="Charakter (Mathematik)">Charakteren von Darstellungen</a> einer Gruppe. Der Name rührt daher, dass man auf einem geeigneten <a href="Funktionenraum" title="Funktionenraum">Funktionenraum</a>, der die Charaktere enthält, ein <a href="Skalarprodukt" title="Skalarprodukt">inneres Produkt</a> definieren kann, bzgl. dessen verschiedene Charaktere tatsächlich orthogonal sind.

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

Im Folgenden sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eine <a href="Endliche_Gruppe" title="Endliche Gruppe">endliche Gruppe</a>. Für einen <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> betrachten wir die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(G,K)}">

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<annotation encoding="application/x-tex">{\displaystyle S(G,K)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be63c521106f8dd862e4ecd9c974500d241b589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.235ex; height:2.843ex;" alt="{\displaystyle S(G,K)}" loading="lazy"> aller Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon G\rightarrow K}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \colon G\rightarrow K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/296cfbda617188c1b9b571b26d068703d7c15a0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.028ex; height:2.176ex;" alt="{\displaystyle \alpha \colon G\rightarrow K}" loading="lazy">. Da man solche Funktionen mittels der Definition

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<annotation encoding="application/x-tex">{\displaystyle (\alpha +k\beta )(g):=\alpha (g)+k\beta (g)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c99820a21ab60de90f03ac8cc22551ba76c8e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.073ex; height:2.843ex;" alt="{\displaystyle (\alpha +k\beta )(g):=\alpha (g)+k\beta (g)}" loading="lazy">   für   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta \in S(G,K),\,k\in K,\,g\in G}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta \in S(G,K),\,k\in K,\,g\in G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3f9cde9521270383f35aee8e2dbe2a3738014b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.673ex; height:2.843ex;" alt="{\displaystyle \alpha ,\beta \in S(G,K),\,k\in K,\,g\in G}" loading="lazy"></dd></dl>

addieren und mit Elementen aus dem Körper multiplizieren kann, liegt ganz offenbar ein <a href="Vektorraum" title="Vektorraum">K-Vektorraum</a> vor. Man kann zwei solche Funktionen sogar multiplizieren, das heißt, es handelt sich sogar um eine <a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper">K-Algebra</a>.

Endlichdimensionale <a href="Darstellung_(Gruppe)" title="Darstellung (Gruppe)">Darstellungen</a> der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> über einem Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> sind <a href="Homomorphismus" title="Homomorphismus">Homomorphismen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \colon G\rightarrow \mathrm {GL} (V)}">

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<annotation encoding="application/x-tex">{\displaystyle \rho \colon G\rightarrow \mathrm {GL} (V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ee666a5572db7125b5e9b589acb595f3f85a902.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.55ex; height:2.843ex;" alt="{\displaystyle \rho \colon G\rightarrow \mathrm {GL} (V)}" loading="lazy"> in die <a href="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeine lineare Gruppe</a> über einem endlichdimensionalen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {tr} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {tr} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/812681cabb4e0ad561bc0e91a24e09d9ae15d6f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.816ex; height:2.009ex;" alt="{\displaystyle \mathrm {tr} }" loading="lazy"> die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} (V)\rightarrow K}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (V)\rightarrow K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3053756e2d0e317cbafab7f0913434e6a2651732.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.554ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (V)\rightarrow K}" loading="lazy">, so nennt man die <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">Komposition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {tr} \circ \rho \colon G\rightarrow K}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {tr} \circ \rho \colon G\rightarrow K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0c52c95bf634de0b8dca708e5c2f5c84bccf1d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.754ex; height:2.676ex;" alt="{\displaystyle \mathrm {tr} \circ \rho \colon G\rightarrow K}" loading="lazy"> den Charakter der Darstellung.

Charaktere von Darstellungen sind offenbar Elemente des Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(G,K)}">

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<annotation encoding="application/x-tex">{\displaystyle S(G,K)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be63c521106f8dd862e4ecd9c974500d241b589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.235ex; height:2.843ex;" alt="{\displaystyle S(G,K)}" loading="lazy">.

Den Charakter einer <a href="Irreduzible_Darstellung" class="mw-redirect" title="Irreduzible Darstellung">irreduziblen Darstellung</a> nennt man ebenfalls irreduzibel.

Wir betrachten von nun an den Fall, dass die <a href="Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> des Körpers kein <a href="Teilbarkeit" title="Teilbarkeit">Teiler</a> der <a href="Gruppenordnung" class="mw-redirect" title="Gruppenordnung">Gruppenordnung</a> ist. Das ist bei Körpern der Charakteristik 0 und damit für die wichtigen Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,\mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27f572434170ccc1369cc16dc5cf71f9ec018b46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.52ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,\mathbb {R} }" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> stets gegeben. Insbesondere können wir im Körper durch die Gruppenordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |G|}">

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<annotation encoding="application/x-tex">{\displaystyle |G|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8258bc41edeb87bfbc8cba8367f29838c0eddc1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.12ex; height:2.843ex;" alt="{\displaystyle |G|}" loading="lazy"> dividieren und damit

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<annotation encoding="application/x-tex">{\displaystyle \langle \alpha ,\beta \rangle _{G}:={\frac {1}{|G|}}\sum _{x\in G}\alpha (x)\beta (x^{-1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20db234015b8497b8bfd9be890283cec39b73def.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.448ex; height:6.509ex;" alt="{\displaystyle \langle \alpha ,\beta \rangle _{G}:={\frac {1}{|G|}}\sum _{x\in G}\alpha (x)\beta (x^{-1})}" loading="lazy"></dd></dl>

definieren. Leicht zeigt man, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \,,\,\rangle _{G}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \langle \,,\,\rangle _{G}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b40ff893e6ac65b5e11aca5142b5d3d97a6e12e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.141ex; height:2.843ex;" alt="{\displaystyle \langle \,,\,\rangle _{G}}" loading="lazy"> eine <a href="Bilinearform" title="Bilinearform">symmetrische, nicht-ausgeartete K-Bilinearform</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(G,K)}">

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<annotation encoding="application/x-tex">{\displaystyle S(G,K)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be63c521106f8dd862e4ecd9c974500d241b589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.235ex; height:2.843ex;" alt="{\displaystyle S(G,K)}" loading="lazy"> ist. Man spricht daher von einem inneren Produkt, auch wenn diese Bezeichnung bei vielen Autoren für die Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> reserviert ist. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58665cdd4df26adaa88a248908d1481041a77c9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {C} }" loading="lazy"> ist wegen der zusätzlichen <a href="Algebraischer_Abschluss" title="Algebraischer Abschluss">algebraischen Abgeschlossenheit</a> der weitaus wichtigste Anwendungsfall.

<div class="mw-heading mw-heading2"><h2 id="Die_Orthogonalitätsrelationen">Die Orthogonalitätsrelationen</h2></div>

Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eine endliche Gruppe, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein Körper, dessen Charakteristik kein Teiler der Gruppenordnung ist, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> seien zwei verschiedene, irreduzible Charaktere der Gruppe über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Dann gilt:<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle \langle \chi ,\psi \rangle _{G}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32f24dd1017f49d79bbed873cf9cc25d493871d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.596ex; height:2.843ex;" alt="{\displaystyle \langle \chi ,\psi \rangle _{G}=0}" loading="lazy">, das heißt, verschiedene irreduzible Charaktere sind orthogonal.</li>

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> algebraisch abgeschlossen, so gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \chi ,\chi \rangle _{G}=1}">

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<annotation encoding="application/x-tex">{\displaystyle \langle \chi ,\chi \rangle _{G}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e94641dfa754cc0730f40a2e49179a5fbadd3c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.538ex; height:2.843ex;" alt="{\displaystyle \langle \chi ,\chi \rangle _{G}=1}" loading="lazy">, das heißt, die irreduziblen Charaktere sind <a href="Orthonormalit%C3%A4t" class="mw-redirect" title="Orthonormalität">orthonormal</a>.</li></ul>

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Charaktertafel" title="Charaktertafel">Charaktertafel</a></div>

Wir betrachten den Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">.

Eine endliche Gruppe hat bekanntlich genau so viele irreduzible Charaktere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{1},\ldots ,\chi _{r}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{1},\ldots ,\chi _{r}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2d852ff74cec726063745e0d740c90d821ad6f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.116ex; height:2.009ex;" alt="{\displaystyle \chi _{1},\ldots ,\chi _{r}}" loading="lazy"> wie <a href="Konjugationsklasse" class="mw-redirect" title="Konjugationsklasse">Konjugationsklassen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1},\ldots ,C_{r}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1},\ldots ,C_{r}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea48991a0db877bf2b03adda419a85fb67aa80bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.53ex; height:2.509ex;" alt="{\displaystyle C_{1},\ldots ,C_{r}}" loading="lazy">. Ferner sind die Charaktere auf Konjugationsklassen konstant, sodass es genügt, die Werte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{i}(c_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{i}(c_{j})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba597b58f8c6366bd6e134ea6c7730da5c222646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.981ex; height:3.009ex;" alt="{\displaystyle \chi _{i}(c_{j})}" loading="lazy"> für beliebig gewählte Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{j}\in C_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{j}\in C_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee8c2ec0018fd9abfb310ad0e1edc5a4288f0c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.329ex; height:2.843ex;" alt="{\displaystyle c_{j}\in C_{j}}" loading="lazy"> zu kennen. Legt man fest, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/babf569931f1a7b5182b9bec51873c2f5692fbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}" loading="lazy"> stets die einelementige Konjugationsklasse des <a href="Neutrales_Element" title="Neutrales Element">neutralen Elements</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b2e3af3644a2872651fa4404c93692ed3821bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{1}}" loading="lazy"> stets der Charakter der trivialen Darstellung sein soll, dann kann man die Gesamtheit der Charaktere leicht in folgendem, Charaktertafel genannten, quadratischen Schema überblicken, wobei die Einträge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}=\chi _{i}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle d_{i}=\chi _{i}(1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/264db085c393ed457de8fbfd7d3317363133f8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.334ex; height:2.843ex;" alt="{\displaystyle d_{i}=\chi _{i}(1)}" loading="lazy"> die Dimensionen der zu den Charakteren gehörigen irreduziblen Darstellungen sind.

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle |C_{2}|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6844e09fa368dac2699b242249ac8745df9989a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.01ex; height:2.843ex;" alt="{\displaystyle |C_{2}|}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle |C_{r}|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79ee7d0de8147bc61fa64f001a03915ff961d2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.929ex; height:2.843ex;" alt="{\displaystyle |C_{r}|}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle c_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b30ba1b247fb8d334580cec68561e749d24aff2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{2}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle c_{r}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70f0d4b75418b6deb56b7a9d66396b3f610e9007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.981ex; height:2.009ex;" alt="{\displaystyle c_{r}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b2e3af3644a2872651fa4404c93692ed3821bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{1}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e06f36f971d825798fbee40aac83239d57bd0f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{2}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9276f8f68c5c23329de74ad76e69f6801358fb1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{2}(c_{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64858d5aea7fddd674ab28ca0b7bfe9135b01ef7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.38ex; height:2.843ex;" alt="{\displaystyle \chi _{2}(c_{2})}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{2}(c_{r})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1824b7ea7c2cfec6ed5a3bf130055f0839beadde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.299ex; height:2.843ex;" alt="{\displaystyle \chi _{2}(c_{r})}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8039d9feb6596ae092e5305108722975060c083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8039d9feb6596ae092e5305108722975060c083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8039d9feb6596ae092e5305108722975060c083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \ddots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f719d9487a2b0e82b38b52421987fedd9de1d4e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.23ex; margin-bottom: -0.401ex; width:2.979ex; height:4.009ex;" alt="{\displaystyle \ddots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8039d9feb6596ae092e5305108722975060c083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{r}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8f81ad4912012ea30e4b916a7f2a3e4ccab51a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.009ex;" alt="{\displaystyle \chi _{r}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle d_{r}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5886109b8e927ee93b63a14330d8fb15895a2cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.183ex; height:2.509ex;" alt="{\displaystyle d_{r}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{r}(c_{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e1955d2e19b9c7c385855218c1d5bb07f5679f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.299ex; height:2.843ex;" alt="{\displaystyle \chi _{r}(c_{2})}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{r}(c_{r})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd19e4e07ce2833488df7ffeb3713cff60ac72d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.219ex; height:2.843ex;" alt="{\displaystyle \chi _{r}(c_{r})}" loading="lazy">

</td></tr>

</tbody></table>

Die Orthogonalitätsrelationen schlagen sich wie folgt in der Charaktertafel nieder.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Orthogonalität_der_Zeilen">Orthogonalität der Zeilen</h3></div>

Die Orthogonalitätsrelationen schreiben sich unter Verwendung des <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Deltas</a> kompakt als

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</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G|\delta _{i,j}=|G|\langle \chi _{i},\chi _{j}\rangle _{G}=\sum _{x\in G}\chi _{i}(x)\chi _{j}(x^{-1})=\sum _{k=0}^{r}|C_{k}|\chi _{i}(c_{k}){\overline {\chi _{j}(c_{k})}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08c0721fb59591a5089b6bfb92a5a9a9f3a77276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.793ex; height:7.009ex;" alt="{\displaystyle |G|\delta _{i,j}=|G|\langle \chi _{i},\chi _{j}\rangle _{G}=\sum _{x\in G}\chi _{i}(x)\chi _{j}(x^{-1})=\sum _{k=0}^{r}|C_{k}|\chi _{i}(c_{k}){\overline {\chi _{j}(c_{k})}}}" loading="lazy">,</dd></dl>

denn die Charaktere sind auf Konjugationsklassen konstant und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (x^{-1})={\overline {\chi (x)}}}">

<msup>

</mrow>

</msup>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi (x^{-1})={\overline {\chi (x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d16a81f8d4aeb398d22c627259bc747c5df921b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.734ex; height:3.676ex;" alt="{\displaystyle \chi (x^{-1})={\overline {\chi (x)}}}" loading="lazy"> für Charaktere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy">.

Trotz der auftretenden Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |C_{k}|}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle |C_{k}|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cff6d190346ca040ced060389a2f32139697a6b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.044ex; height:2.843ex;" alt="{\displaystyle |C_{k}|}" loading="lazy"> nennt man diese Beziehung die Orthogonalität der Zeilen der Charaktertafel. Man kann diese Gleichungen auch als <a href="Matrizenmultiplikation" title="Matrizenmultiplikation">Matrizenmultiplikation</a> lesen. Definiert man nämlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=(\chi _{i}(c_{k}))_{i,k=1,\ldots ,n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=(\chi _{i}(c_{k}))_{i,k=1,\ldots ,n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/303101ac5c44d6e2cb47832676ec90f5de7dc68a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.088ex; height:3.009ex;" alt="{\displaystyle X=(\chi _{i}(c_{k}))_{i,k=1,\ldots ,n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=(|C_{k}|{\overline {\chi _{j}(c_{k})}})_{k,j=1,\ldots ,n}}">

</mrow>

<msub>

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</msub>

</mrow>

<mover>

<mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mrow>

</mover>

</mrow>

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=(|C_{k}|{\overline {\chi _{j}(c_{k})}})_{k,j=1,\ldots ,n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6875933bae6ba90391e9d7a5f179bdc8a515e718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.261ex; height:3.843ex;" alt="{\displaystyle Y=(|C_{k}|{\overline {\chi _{j}(c_{k})}})_{k,j=1,\ldots ,n}}" loading="lazy">, so ist obige Gleichung nichts anderes als

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G|\cdot 1_{r}=XY}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64cee62160c374b0866225fa629be71d1b072243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.787ex; height:2.843ex;" alt="{\displaystyle |G|\cdot 1_{r}=XY}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{r}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{r}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3052e224f8f2c76b474bcfaaf672a057dde57956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.136ex; height:2.509ex;" alt="{\displaystyle 1_{r}}" loading="lazy"> die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a> ist. Insbesondere sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> <a href="Invertierbare_Matrix" class="mw-redirect" title="Invertierbare Matrix">invertierbar</a>.

<div class="mw-heading mw-heading3"><h3 id="Orthogonalität_der_Spalten">Orthogonalität der Spalten</h3></div>

Multipliziert man obige Matrixgleichung von links mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71f75a18781a402f9fb09972f99e092bf21aa877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.33ex; height:2.676ex;" alt="{\displaystyle X^{-1}}" loading="lazy"> und von rechts mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, so erhält man:

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G|\cdot 1_{r}=YX}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1706ea0e5f3c78f25cf77aa49f49938692bec8a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.787ex; height:2.843ex;" alt="{\displaystyle |G|\cdot 1_{r}=YX}" loading="lazy"></dd></dl>

In Komponentenschreibweise bedeutet das

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G|\delta _{k,l}=\sum _{j=1}^{r}|C_{k}|{\overline {\chi _{j}(c_{k})}}\chi _{j}(c_{l})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d68b65e2e36228dc85f429f0b48af47c36b15568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.362ex; height:7.176ex;" alt="{\displaystyle |G|\delta _{k,l}=\sum _{j=1}^{r}|C_{k}|{\overline {\chi _{j}(c_{k})}}\chi _{j}(c_{l})}" loading="lazy"></dd></dl>

oder, da das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |C_{k}|}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |C_{k}|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cff6d190346ca040ced060389a2f32139697a6b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.044ex; height:2.843ex;" alt="{\displaystyle |C_{k}|}" loading="lazy"> unter der Summe konstant ist:

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mover>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {|G|}{|C_{k}|}}\delta _{k,l}=\sum _{j=1}^{r}{\overline {\chi _{j}(c_{k})}}\chi _{j}(c_{l})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9fe6590eef77e5186fcb60f73a1ff18a2f0fd92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.078ex; height:7.176ex;" alt="{\displaystyle {\frac {|G|}{|C_{k}|}}\delta _{k,l}=\sum _{j=1}^{r}{\overline {\chi _{j}(c_{k})}}\chi _{j}(c_{l})}" loading="lazy"></dd></dl>

Diese Beziehung nennt man in naheliegender Weise die Orthogonalität der Spalten.

<div class="mw-heading mw-heading2"><h2 id="Orthogonalitätsrelationen_für_Darstellungen">Orthogonalitätsrelationen für Darstellungen</h2></div>

Da Charaktere die Spuren von Darstellungen sind, wird man ähnliche Orthogonalitätsrelationen für Darstellungen erwarten, tatsächlich werden diese für den Beweis obiger Orthogonalitätsrelationen verwendet. Da Darstellungen ihre Werte aber nicht im Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, sondern in allgemeinen linearen Gruppen über Vektorräumen annehmen, ist die Formulierung etwas aufwändiger. Wie schon oben beschränken wir uns auf endlichdimensionale Darstellungen und wählen als Vektorraum einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">-dimensionalen Darstellung den <a href="Koordinatenraum" title="Koordinatenraum">Koordinatenraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{d}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{d}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba24fc3a254608fbde73fffa05104d5e3a98dea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.186ex; height:2.676ex;" alt="{\displaystyle K^{d}}" loading="lazy">, was letztlich der nicht eindeutigen Wahl einer Basis entspricht. Eine Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \colon G\rightarrow \mathrm {GL} (K^{d})\cong \mathrm {Mat} (d,K)}">

</mrow>

<msup>

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</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \colon G\rightarrow \mathrm {GL} (K^{d})\cong \mathrm {Mat} (d,K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcc374db0e8f9e4f0e0418a94eabb3894769ba6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.371ex; height:3.176ex;" alt="{\displaystyle \rho \colon G\rightarrow \mathrm {GL} (K^{d})\cong \mathrm {Mat} (d,K)}" loading="lazy"> hat damit Werte in den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">-reihigen quadratischen Matrizen über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und man kann die Komponentenfunktionen

<msub>

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</msub>

<msub>

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</msub>

<mtext>(i,j)-te Komponente von </mtext>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{i,j}\colon G\rightarrow K,\quad \rho _{i,j}(x)={\text{(i,j)-te Komponente von }}\rho (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c8b1591b557503268971c18755be48f25fecdb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.615ex; height:3.009ex;" alt="{\displaystyle \rho _{i,j}\colon G\rightarrow K,\quad \rho _{i,j}(x)={\text{(i,j)-te Komponente von }}\rho (x)}" loading="lazy"></dd></dl>

betrachten. Mit diesen Definitionen besteht folgender Satz:<a href="#cite_note-4">[4]</a>

Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eine endliche Gruppe, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein Körper, dessen Charakteristik kein Teiler der Gruppenordnung ist; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> seien zwei irreduzible Darstellungen der Gruppe über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Dann gilt:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> nicht äquivalent, so ist</li></ul>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{x\in G}\rho _{i,j}(x)\sigma _{r,s}(x^{-1})=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f8273515cf78fa6bb7cf987a690e4d5fd6ff61b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.28ex; height:5.676ex;" alt="{\displaystyle \sum _{x\in G}\rho _{i,j}(x)\sigma _{r,s}(x^{-1})=0}" loading="lazy">   für alle Komponentenfunktionen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">.</dd></dl></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> algebraisch abgeschlossen, so gilt</li></ul>

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</munder>

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<msup>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{x\in G}\rho _{i,j}(x)\rho _{r,s}(x^{-1})={\frac {|G|}{\mathrm {dim} (\rho )}}\delta _{i,s}\delta _{j,r}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca84c3f4ae654a4dd8662290c0b0b27b64f46199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.916ex; height:7.009ex;" alt="{\displaystyle \sum _{x\in G}\rho _{i,j}(x)\rho _{r,s}(x^{-1})={\frac {|G|}{\mathrm {dim} (\rho )}}\delta _{i,s}\delta _{j,r}}" loading="lazy">   für alle Komponentenfunktionen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">.</dd></dl></dd></dl>

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

Die Orthogonalitätsrelationen bilden einen Eckpfeiler der sehr weit ausgebauten <a href="Darstellungstheorie_endlicher_Gruppen" title="Darstellungstheorie endlicher Gruppen">Darstellungstheorie der Gruppen</a>. Wir beschränken uns im Folgenden auf den Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58665cdd4df26adaa88a248908d1481041a77c9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {C} }" loading="lazy"> und bringen nur einige sehr elementare Anwendungen, um den Einsatz der Orthogonalitätsrelationen zu verdeutlichen.

<div class="mw-heading mw-heading3"><h3 id="Summen_irreduzibler_Charaktere">Summen irreduzibler Charaktere</h3></div>

Die verschiedenen, irreduziblen Charaktere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{1},\ldots ,\chi _{r}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{1},\ldots ,\chi _{r}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2d852ff74cec726063745e0d740c90d821ad6f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.116ex; height:2.009ex;" alt="{\displaystyle \chi _{1},\ldots ,\chi _{r}}" loading="lazy"> einer Gruppe sind nicht nur orthonormal, aus Dimensionsgründen erzeugen sie auch den Raum der sogenannten Klassenfunktionen, das heißt von Funktionen, die auf Konjugationsklassen konstant sind. Die irreduziblen Charaktere bilden daher eine Orthonormalbasis im Raum der Klassenfunktionen. Insbesondere ist jede Klassenfunktion eine eindeutige Linearkombination irreduzibler Charaktere.<a href="#cite_note-5">[5]</a>

Nach dem <a href="Satz_von_Maschke" title="Satz von Maschke">Satz von Maschke</a> ist jede endlichdimensionale Darstellung einer endlichen Gruppe direkte Summe irreduzibler Darstellungen. Durch Spurbildung erhält man, dass jeder Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> Summe irreduzibler Charaktere ist, das heißt:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi =\sum _{i=1}^{r}n_{i}\chi _{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7553806ad8df0ba08403a5b4b61e6ddcc767b15b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.745ex; height:6.843ex;" alt="{\displaystyle \chi =\sum _{i=1}^{r}n_{i}\chi _{i}}" loading="lazy">   mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}\in \mathbb {N} _{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}\in \mathbb {N} _{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b91c5470c56f357ee32c980f0f3d6bf4ed1b160a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.767ex; height:2.509ex;" alt="{\displaystyle n_{i}\in \mathbb {N} _{0}}" loading="lazy"></dd></dl>

Die Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy"> lassen sich mittels Orthogonalität sofort bestimmen:

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</munderover>

<msub>

</mrow>

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<msub>

</mrow>

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</munderover>

<msub>

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<msub>

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<msub>

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<msub>

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</munderover>

<msub>

</mrow>

</msub>

<msub>

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<msub>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}=\sum _{j=1}^{r}\delta _{i,j}n_{j}=\sum _{j=1}^{r}n_{j}\langle \chi _{i},\chi _{j}\rangle _{G}=\langle \chi _{i},\sum _{j=1}^{r}n_{j}\chi _{j}\rangle _{G}=\langle \chi _{i},\chi \rangle _{G}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53317bfb7c1fd6c6f25b5e67bb91771d5c4638a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:61.745ex; height:7.176ex;" alt="{\displaystyle n_{i}=\sum _{j=1}^{r}\delta _{i,j}n_{j}=\sum _{j=1}^{r}n_{j}\langle \chi _{i},\chi _{j}\rangle _{G}=\langle \chi _{i},\sum _{j=1}^{r}n_{j}\chi _{j}\rangle _{G}=\langle \chi _{i},\chi \rangle _{G}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Irreduzibilitätkriterium">Irreduzibilitätkriterium</h3></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> eine endlichdimensionale Darstellung mit Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> genau dann irreduzibel, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \chi ,\chi \rangle _{G}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \chi ,\chi \rangle _{G}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e94641dfa754cc0730f40a2e49179a5fbadd3c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.538ex; height:2.843ex;" alt="{\displaystyle \langle \chi ,\chi \rangle _{G}=1}" loading="lazy">.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>

Beweis: Dass Charaktere irreduzibler Darstellungen diese Eigenschaft haben, ist der zweite Punkt obiger Orthogonalitätsrelationen. Umgekehrt ist jeder Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi =\sum _{i=1}^{r}n_{i}\chi _{i}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi =\sum _{i=1}^{r}n_{i}\chi _{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a6d3390d8bce47568b9b2615d179c92491688a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.744ex; height:3.176ex;" alt="{\displaystyle \textstyle \chi =\sum _{i=1}^{r}n_{i}\chi _{i}}" loading="lazy"> Summe irreduzibler Charaktere und daraus folgt wegen der Orthonormalität <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \langle \chi ,\chi \rangle _{G}=\sum _{i=1}^{r}n_{i}^{2}}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \langle \chi ,\chi \rangle _{G}=\sum _{i=1}^{r}n_{i}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0fbc499d4402eb5f824994c41b3c1b3a9f2c048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.566ex; height:3.176ex;" alt="{\displaystyle \textstyle \langle \chi ,\chi \rangle _{G}=\sum _{i=1}^{r}n_{i}^{2}}" loading="lazy"> mit natürlichen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy">. Ist dies gleich 1, so bleibt nur die Möglichkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92e3cf1197d752e9005b9b131133f07bb3885ce6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.455ex; height:2.509ex;" alt="{\displaystyle n_{i}=1}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{j}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{j}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfcd1ec45c2606a560b211d13c4d18a797de4557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.565ex; height:2.843ex;" alt="{\displaystyle n_{j}=0}" loading="lazy"> für alle anderen Koeffizienten. Daraus folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi =\chi _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi =\chi _{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0526058b5e54cae5e0c7e9e95fb4a016ae898c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.808ex; height:2.009ex;" alt="{\displaystyle \chi =\chi _{i}}" loading="lazy">, das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> ist irreduzibel und damit auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Vervollständigung_von_Charaktertafeln">Vervollständigung von Charaktertafeln</h3></div>

Mittels der Orthogonalitätsrelationen können Teile von Charaktertafeln erschlossen werden.

Als Beispiel betrachten wir die <a href="S3_(Gruppe)" title="S3 (Gruppe)">symmetrische Gruppe S3</a>. Neben der trivialen Konjugationsklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/babf569931f1a7b5182b9bec51873c2f5692fbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}" loading="lazy"> haben wir die Konjugationsklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ec545f7870665e1028b7492746848d149878808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}" loading="lazy"> der drei <a href="Zyklische_Permutation" title="Zyklische Permutation">Transpositionen</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{3}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle C_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66e9abeb5057b7afbf88e3169101849354f13c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{3}}" loading="lazy"> der beiden Elemente der <a href="Ordnung_eines_Gruppenelementes" title="Ordnung eines Gruppenelementes">Ordnung</a> 3. Als offensichtliche eindimensionale Darstellungen haben wir die triviale Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b2e3af3644a2872651fa4404c93692ed3821bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{1}}" loading="lazy"> und die <a href="Vorzeichen_(Permutation)" title="Vorzeichen (Permutation)">Signum-Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e06f36f971d825798fbee40aac83239d57bd0f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{2}}" loading="lazy">. Da es genau so viele Charaktere wie Konjugationsklassen gibt, fehlt noch ein Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a81078ba5663b5250c23e9c14d4caf4d19123025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{3}}" loading="lazy">, dessen Werte wir noch nicht kennen. Die Charaktertafel hat also die Gestalt

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70e15f3e200aaa247f69c43110cc5a09ecc91b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy">

</th></tr>

<tr>

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11c8046c70c7e0e84425b052f781db3fc6dc4cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.322ex; height:2.509ex;" alt="{\displaystyle c_{1}=1}" loading="lazy">

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{2}=(1,2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70e66fead0ca60b95aff8560fd046d04e1fe45cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.328ex; height:2.843ex;" alt="{\displaystyle c_{2}=(1,2)}" loading="lazy">

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{3}=(1,2,3)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ffb3014028268e14dd1fcaadf6c96ed1643b4aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.524ex; height:2.843ex;" alt="{\displaystyle c_{3}=(1,2,3)}" loading="lazy">

</th></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b2e3af3644a2872651fa4404c93692ed3821bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{1}}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e06f36f971d825798fbee40aac83239d57bd0f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{2}}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a81078ba5663b5250c23e9c14d4caf4d19123025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{3}}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">

</td></tr>

</tbody></table>

mit noch unbekannten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbeca34b28f569a407ef74a955d041df9f360268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}" loading="lazy">. Diese lassen sich mittels der Orthogonalitätsrelationen bestimmen, ohne die fehlende irreduzible Darstellung zu kennen, es werden nicht einmal weitere Details der Gruppe benötigt.

Aus der Orthogonalität für Spalten folgt für die erste Spalte

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 6=|S_{3}|=1^{2}+1^{2}+x^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d4161817d9f7a87e34f3f2135c3905186269205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.63ex; height:3.176ex;" alt="{\displaystyle 6=|S_{3}|=1^{2}+1^{2}+x^{2}}" loading="lazy">,</dd></dl>

also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=\pm 2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/108e0bbaa344d385e95ac7172e8a891c4fbf430a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle x=\pm 2}" loading="lazy">. Da in der ersten Spalte aber die Dimensionen (Spuren von Einheitsmatrizen) stehen, muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy"> sein, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}" loading="lazy">.

Für die zweite Spalte folgt

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {6}{3}}={\frac {|S_{3}|}{|C_{2}|}}=1^{2}+1^{2}+y^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/611852877dc4bd7c6775b9d4c321a11c9c79291e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.37ex; height:6.509ex;" alt="{\displaystyle {\frac {6}{3}}={\frac {|S_{3}|}{|C_{2}|}}=1^{2}+1^{2}+y^{2}}" loading="lazy">,</dd></dl>

und da bleibt nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/094f824655138f6b11d96a0da32e7f0716ba6959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}" loading="lazy">.

Da die dritte Spalte zur ersten orthogonal ist, folgt

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0=1\cdot 1+1\cdot 1+2\cdot z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab46c738c42b50430cc4e018cfff63ac98a0b133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.879ex; height:2.343ex;" alt="{\displaystyle 0=1\cdot 1+1\cdot 1+2\cdot z}" loading="lazy">,</dd></dl>

also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af5d92c041c1ddaa688b8f7f68d333e107e93709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.157ex; height:2.343ex;" alt="{\displaystyle z=-1}" loading="lazy">. Damit ist die Charaktertafel der Gruppe S3 vollständig bestimmt.<a href="#cite_note-8">[8]</a>

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

Die Orthogonalitätsrelationen gehen auf eine Arbeit von <a href="Ferdinand_Georg_Frobenius" title="Ferdinand Georg Frobenius">Ferdinand Georg Frobenius</a> aus dem Jahre 1896 zurück, dort werden auch Charaktertafeln besprochen.<a href="#cite_note-9">[9]</a> Eine Überarbeitung dieser Theorie wurde von <a href="Issai_Schur" title="Issai Schur">Issai Schur</a> unternommen,<a href="#cite_note-10">[10]</a> man findet daher auch die Bezeichnung schursche Orthogonalitätsrelationen.

Endliche Gruppen sind <a href="Lokalkompakte_Gruppe" title="Lokalkompakte Gruppe">kompakte Gruppen</a>, deren <a href="Haarsches_Ma%C3%9F" title="Haarsches Maß">haarsches Maß</a> jeder einelementigen Menge das Maß <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{|G|}}}">

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{|G|}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2de5f33469f151e9a875cf588fda4e0adfaa9be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.043ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {1}{|G|}}}" loading="lazy"> zuordnet. Man erhält analoge Resultate für unendliche kompakte Gruppen, wenn man Summationen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{|G|}}\sum _{x\in G}}">

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{|G|}}\sum _{x\in G}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/326fed86988f3cc4700c171c2f5e7f80295b8916.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.444ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {1}{|G|}}\sum _{x\in G}}" loading="lazy"> durch Integrale nach dem haarschen Maß ersetzt. <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> hatte 1934 erste Ergebnisse in dieser Richtung erzielt, allerdings noch unter Verwendung <a href="Fastperiodische_Funktion" title="Fastperiodische Funktion">fastperiodischer Funktionen</a>.<a href="#cite_note-11">[11]</a> Eine modernere Darstellung, die das haarsche Maß verwendet, findet sich zum Beispiel im unten genannten Lehrbuch „Representations of Finite and Compact Groups“ von Barry Simon.<a href="#cite_note-12">[12]</a>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Derek_John_Scott_Robinson" title="Derek John Scott Robinson">D. J. S. Robinson</a>: A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3, Satz 8.3.5.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2, Satz 9.5.6 (Orthogonalitätsrelation).

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> D. J. S. Robinson: A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3, Kapitel 8, Seite 232: The Character Table.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> D. J. S. Robinson: A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3, Satz 8.3.4.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2, Satz 9.6.6.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> D. J. S. Robinson: A Course in the Theory of Groups. Springer-Verlag, 1996, ISBN 0-387-94461-3, Satz 8.3.12.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2, Satz 9.6.4.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2, Beispiel 9.7.1.b.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> F. G. Frobenius: Über Gruppencharktere. Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin 1896, <a rel="nofollow" class="external text" href="https://www.e-rara.ch/doi/10.3931/e-rara-18877">hier als PDF erhältlich.</a>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> I. Schur: Neue Begründung der Theorie der Gruppencharaktere. Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin 1905, Seite 406.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> J. v. Neumann: Almost periodic functions in groups. Trans. Amer. Math. Soc. (1934), Band 36, Nr. 3, Seiten 445–492.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> Barry Simon: Representations of Finite and Compact Groups. American Mathematical Society 1996, Band 10.

</li>

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