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<title>Invariantes Polynom</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Invariantes Polynom</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">In der <a href="Mathematik" title="Mathematik">Mathematik</a> ist ein invariantes Polynom ein <a href="Polynom" title="Polynom">Polynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> auf einem Vektorraum (siehe <a href="Symmetrische_Algebra" title="Symmetrische Algebra">Symmetrische Algebra</a>), welches unter der Wirkung einer <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> <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"> auf dem <a href="Vektorraum" title="Vektorraum">Vektorraum</a> <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"> invariant ist, also

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<annotation encoding="application/x-tex">{\displaystyle P(gx)=P(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2967448b18ce44844fe528572bd539a97bf6c83d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.983ex; height:2.843ex;" alt="{\displaystyle P(gx)=P(x)}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G,x\in V}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/536d8dc2995eb9c2a979c3cea80e882fd7e62fa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.775ex; height:2.509ex;" alt="{\displaystyle g\in G,x\in V}" loading="lazy"> erfüllt.

<div class="mw-heading mw-heading2"><h2 id="Invariante_Polynome_in_der_Linearen_Algebra">Invariante Polynome in der Linearen Algebra</h2></div>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy"> ein <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\operatorname {Mat} (n,\mathbb {K} )}">

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<annotation encoding="application/x-tex">{\displaystyle V=\operatorname {Mat} (n,\mathbb {K} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2a145ce047af8c093a5fe76b9cf8a5592f514c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.13ex; height:2.843ex;" alt="{\displaystyle V=\operatorname {Mat} (n,\mathbb {K} )}" loading="lazy"> der Vektorraum aller <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Matrizen über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy">. Die <a href="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeine lineare Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {K} )}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {K} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdb04a667a7b8e6e5d36673ad9ba00415c529651.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.323ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {K} )}" loading="lazy"> wirkt auf <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"> durch Konjugation:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1d4622fcde9c2614504de6dfc68d04fca26811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.088ex; height:3.009ex;" alt="{\displaystyle gx:=gxg^{-1}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in \operatorname {GL} (n,\mathbb {K} ),x\in \operatorname {Mat} (n,\mathbb {K} )}">

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<annotation encoding="application/x-tex">{\displaystyle g\in \operatorname {GL} (n,\mathbb {K} ),x\in \operatorname {Mat} (n,\mathbb {K} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a1dfaaab282b0c6d72f5ffacba8b2ab6582040a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.728ex; height:2.843ex;" alt="{\displaystyle g\in \operatorname {GL} (n,\mathbb {K} ),x\in \operatorname {Mat} (n,\mathbb {K} )}" loading="lazy">.</dd></dl>

Invariante Polynome sind in diesem Fall Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:\operatorname {Mat} (n,\mathbb {K} )\rightarrow \mathbb {K} }">

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<annotation encoding="application/x-tex">{\displaystyle P:\operatorname {Mat} (n,\mathbb {K} )\rightarrow \mathbb {K} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4222b5084e0b69b26381069c17815b6060918f61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.349ex; height:2.843ex;" alt="{\displaystyle P:\operatorname {Mat} (n,\mathbb {K} )\rightarrow \mathbb {K} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(gxg^{-1})=P(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd7f33e8eac1e0acc2d3b8fa1e3ae4d1ecb29bb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.434ex; height:3.176ex;" alt="{\displaystyle P(gxg^{-1})=P(x)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in \operatorname {GL} (n,\mathbb {K} ),x\in \operatorname {Mat} (n,\mathbb {K} )}">

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<annotation encoding="application/x-tex">{\displaystyle g\in \operatorname {GL} (n,\mathbb {K} ),x\in \operatorname {Mat} (n,\mathbb {K} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a1dfaaab282b0c6d72f5ffacba8b2ab6582040a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.728ex; height:2.843ex;" alt="{\displaystyle g\in \operatorname {GL} (n,\mathbb {K} ),x\in \operatorname {Mat} (n,\mathbb {K} )}" loading="lazy">.

Beispiele sind die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> und die <a href="Determinante" title="Determinante">Determinante</a> von Matrizen. Allgemeiner kann man (mit einer formalen Variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">) die Entwicklung

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<annotation encoding="application/x-tex">{\displaystyle \det(tA+I)=\sum _{k=0}^{n}c_{k}(A)t^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52a888da34435dd7b5f21e756097e4b77d861d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.051ex; height:7.009ex;" alt="{\displaystyle \det(tA+I)=\sum _{k=0}^{n}c_{k}(A)t^{k}}" loading="lazy"></dd></dl>

betrachten und erhält invariante Polynome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0},\ldots ,c_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f51a842a002211594d9cf0bebffea87c951e9f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.465ex; height:2.009ex;" alt="{\displaystyle c_{0},\ldots ,c_{n}}" loading="lazy">. (<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_/eb734a37dd21ce173a46342d1cc64c92/77b7dc6d279091d354e0b90889b463bfa7eb7247.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_{1}}" loading="lazy"> ist die Spur und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b7e944bcb1be88e9a6a940638f2adce0ec4211a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.009ex;" alt="{\displaystyle c_{n}}" loading="lazy"> die Determinante. Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1848c435e64864e9ad4efa7e46bd6bc900c35c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }" loading="lazy"> <a href="Algebraischer_Abschluss" title="Algebraischer Abschluss">algebraisch abgeschlossen</a> ist, dann ist allgemein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d2f8052630e67b00d04e3487e1d68ed7070470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.096ex; height:2.009ex;" alt="{\displaystyle c_{k}}" loading="lazy"> das k-te <a href="Symmetrisches_Polynom#Elementarsymmetrische_Polynome" title="Symmetrisches Polynom">elementarsymmetrische Polynom</a> in den <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerten</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">.)

<div class="mw-heading mw-heading2"><h2 id="Invariante_Polynome_in_der_Theorie_der_Lie-Gruppen">Invariante Polynome in der Theorie der Lie-Gruppen</h2></div>

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="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> ihre <a href="Lie-Gruppe#Lie-Algebra_der_Lie-Gruppe" title="Lie-Gruppe">Lie-Algebra</a>. Ein Polynom auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> ist ein Polynom (mit reellen Koeffizienten) in den Basisvektoren von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">, siehe <a href="Symmetrische_Algebra" title="Symmetrische Algebra">Symmetrische Algebra</a>.

Die 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"> wirkt auf sich selbst durch Konjugation:

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<annotation encoding="application/x-tex">{\displaystyle c_{g}(h):=ghg^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b9c7a8047fb3c373b8ff44ab7877823605b6713.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.828ex; height:3.343ex;" alt="{\displaystyle c_{g}(h):=ghg^{-1}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in G}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c938e8bf0bc7fa38baf00c88fa8aa6acf633f823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle h\in G}" loading="lazy">.

Das Differential von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{g}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f66b66841d4193e6ef61fe348c322d7616acc32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.028ex; height:2.343ex;" alt="{\displaystyle c_{g}}" loading="lazy"> ist eine lineare Abbildung

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ad} (g):=D(c_{g})_{e}:{\mathfrak {g}}\rightarrow {\mathfrak {g}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c5739c3d881dc836d47a08dace69f796c967c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.361ex; height:3.009ex;" alt="{\displaystyle \operatorname {Ad} (g):=D(c_{g})_{e}:{\mathfrak {g}}\rightarrow {\mathfrak {g}}}" loading="lazy">,</dd></dl>

dies definiert die sogenannte <a href="Adjungierte_Darstellung" title="Adjungierte Darstellung">adjungierte Darstellung</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>

</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"> auf dem Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">.

Ein invariantes Polynom ist ein Polynom auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">, welches invariant unter der adjungierten Wirkung ist, also

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\operatorname {Ad} (g)X_{1},\ldots ,\operatorname {Ad} (g)X_{k})=P(X_{1},\ldots ,X_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc6c0798e5492e39910ec95d5870f6e08eb842f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.469ex; height:2.843ex;" alt="{\displaystyle P(\operatorname {Ad} (g)X_{1},\ldots ,\operatorname {Ad} (g)X_{k})=P(X_{1},\ldots ,X_{k})}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G,X_{1},\ldots ,X_{k}\in {\mathfrak {g}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G,X_{1},\ldots ,X_{k}\in {\mathfrak {g}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/312560af8db925797d93e095be5842796c668da9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.999ex; height:2.509ex;" alt="{\displaystyle g\in G,X_{1},\ldots ,X_{k}\in {\mathfrak {g}}}" loading="lazy"></dd></dl>

erfüllt. Die Algebra der invarianten Polynome wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I^{*}({\mathfrak {g}})}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I^{*}({\mathfrak {g}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77b5eb0ef30a39344ad7d8400831f22ddf9d80c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.252ex; height:2.843ex;" alt="{\displaystyle I^{*}({\mathfrak {g}})}" loading="lazy"> bezeichnet.

<div class="mw-heading mw-heading3"><h3 id="Beispiel_G=GL(n,ℝ)">Beispiel G=GL(n,ℝ)</h3></div>

In diesem Fall ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}=\operatorname {Mat} (n,\mathbb {R} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}=\operatorname {Mat} (n,\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/072b266f7f44d9f868f49f7112a482dfc2e833c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.384ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}=\operatorname {Mat} (n,\mathbb {R} )}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ad(g)(A)=gAg^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Ad(g)(A)=gAg^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e5687ddb8c88e97a5f807ab5ed0fdd2ead142f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.845ex; height:3.176ex;" alt="{\displaystyle Ad(g)(A)=gAg^{-1}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in \operatorname {GL} (n,\mathbb {R} ),A\in \operatorname {Mat} (n,\mathbb {R} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in \operatorname {GL} (n,\mathbb {R} ),A\in \operatorname {Mat} (n,\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c295e8c18cdae5a8fb7ee0a691e760c584b819d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.882ex; height:2.843ex;" alt="{\displaystyle g\in \operatorname {GL} (n,\mathbb {R} ),A\in \operatorname {Mat} (n,\mathbb {R} )}" loading="lazy">. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy"> sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\frac {k}{2}}}">

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66490dfdb0017171010adbb6c0a966c349dfd6c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.256ex; height:4.009ex;" alt="{\displaystyle P_{\frac {k}{2}}}" loading="lazy"> das homogene Polynom vom Grad <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/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, dessen Wert auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,\ldots ,A)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc991cf6f116db66cdeccf44da9c20b2a8bcea78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.474ex; height:2.843ex;" alt="{\displaystyle (A,\ldots ,A)}" loading="lazy"> man als Koeffizienten vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy"> im Polynom

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi }}A\right)=\sum _{k}P_{\frac {k}{2}}(A,\ldots ,A)\lambda ^{n-k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9383b8ea3573eccfe582746058aeae746521c49e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.491ex; height:6.676ex;" alt="{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi }}A\right)=\sum _{k}P_{\frac {k}{2}}(A,\ldots ,A)\lambda ^{n-k}}" loading="lazy"></dd></dl>

erhält, für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \operatorname {Mat} (n,\mathbb {R} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (n,\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/985427e2fd410e124071c94da959debbd9431a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.698ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (n,\mathbb {R} )}" loading="lazy">. (Die Werte für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,\ldots ,A)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc991cf6f116db66cdeccf44da9c20b2a8bcea78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.474ex; height:2.843ex;" alt="{\displaystyle (A,\ldots ,A)}" loading="lazy"> legen ein Polynom bereits eindeutig fest.) Das Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\frac {k}{2}}}">

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66490dfdb0017171010adbb6c0a966c349dfd6c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.256ex; height:4.009ex;" alt="{\displaystyle P_{\frac {k}{2}}}" loading="lazy"> heißt das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {k}{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84200f32cf81d1c85e1444ffb36c2c4f3ec84545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.047ex; height:5.343ex;" alt="{\displaystyle {\frac {k}{2}}}" loading="lazy">-te Pontrjagin-Polynom.

Die Algebra der invarianten Polynome wird von den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\frac {k}{2}}\in I^{k}({\mathfrak {gl}}(n,\mathbb {R} ))}">

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\frac {k}{2}}\in I^{k}({\mathfrak {gl}}(n,\mathbb {R} ))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73c355dc51850e95d691ef29dc2064c74b2b095b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.95ex; height:4.509ex;" alt="{\displaystyle P_{\frac {k}{2}}\in I^{k}({\mathfrak {gl}}(n,\mathbb {R} ))}" loading="lazy"> erzeugt.

<div class="mw-heading mw-heading3"><h3 id="Beispiel_G=O(n)">Beispiel G=O(n)</h3></div>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathfrak {o}}(n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathfrak {o}}(n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4040985c5e30a0ecd5316875658e14fe2bf42b36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.925ex; height:2.843ex;" alt="{\displaystyle A\in {\mathfrak {o}}(n)}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=-A^{T}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61cb347027106880a8d192099b51fb9bd70551ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.782ex; height:2.843ex;" alt="{\displaystyle A=-A^{T}}" loading="lazy">, woraus zunächst <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi }}A\right)=\det \left(\lambda \mathbb {I} +{\frac {1}{2\pi }}A\right)}">

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi }}A\right)=\det \left(\lambda \mathbb {I} +{\frac {1}{2\pi }}A\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bb9b9c785e877baf7ac2dc4b9eec6a74e451b76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.522ex; height:6.176ex;" alt="{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi }}A\right)=\det \left(\lambda \mathbb {I} +{\frac {1}{2\pi }}A\right)}" loading="lazy"> und damit dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\frac {k}{2}}=0}">

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\frac {k}{2}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3e3bf26c7ba2008665a5c06bc6d16e1bb4edb5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.517ex; height:4.009ex;" alt="{\displaystyle P_{\frac {k}{2}}=0}" loading="lazy"> für alle ungeraden <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/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> folgt.

Die Algebra der invarianten Polynome wird von den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{k}\in I^{2k}({\mathfrak {o}}(n))}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{k}\in I^{2k}({\mathfrak {o}}(n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7a071325e7a7061b270a0381fb548cc1fc83fb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.699ex; height:3.176ex;" alt="{\displaystyle P_{k}\in I^{2k}({\mathfrak {o}}(n))}" loading="lazy"> erzeugt.

<div class="mw-heading mw-heading3"><h3 id="Beispiel_G=SO(n)">Beispiel G=SO(n)</h3></div>

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d42d6f855a8c9c526b0bebd6a41389ca605c36c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.696ex; height:2.176ex;" alt="{\displaystyle n=2m}" loading="lazy"> gerade ist, hat man zusätzlich noch die <a href="Pfaffsche_Determinante" title="Pfaffsche Determinante">Pfaffsche Determinante</a>, die für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(a_{ij})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=(a_{ij})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/296ad42d9541f8285979ce822ccb661da56111ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.358ex; height:3.009ex;" alt="{\displaystyle A=(a_{ij})}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}=-a_{ji}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd578e51eb45764d11cbdcb5fdcfecec755f36f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.32ex; height:2.676ex;" alt="{\displaystyle a_{ij}=-a_{ji}}" loading="lazy"> definiert ist durch

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Pf(A,\ldots ,A)={\frac {1}{2^{2m}\pi ^{m}m!}}\sum _{\sigma \in S_{2m}}sign(\sigma )a_{\sigma (1)\sigma (2)}\ldots a_{\sigma (2m-1)\sigma (2m)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/437edec32dbcf15ba9b8893a0e18de8e9d401e4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:63.726ex; height:6.676ex;" alt="{\displaystyle Pf(A,\ldots ,A)={\frac {1}{2^{2m}\pi ^{m}m!}}\sum _{\sigma \in S_{2m}}sign(\sigma )a_{\sigma (1)\sigma (2)}\ldots a_{\sigma (2m-1)\sigma (2m)}}" loading="lazy">.</dd></dl>

Die Algebra der invarianten Polynome wird von den Pontrjagin-Polynomen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{k}\in I^{2k}({\mathfrak {so}}(n))}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{k}\in I^{2k}({\mathfrak {so}}(n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc7734a8c7ef889ba4fe7635578117caeee3fff4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.729ex; height:3.176ex;" alt="{\displaystyle P_{k}\in I^{2k}({\mathfrak {so}}(n))}" loading="lazy"> und – falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> gerade ist – der (auch als Euler-Polynom bezeichneten) Pfaffschen Determinante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Pf\in I^{\frac {n}{2}}({\mathfrak {so}}(n))}">

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Pf\in I^{\frac {n}{2}}({\mathfrak {so}}(n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31f3198804b42663e6c02a53b7748f774962912b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.13ex; height:3.676ex;" alt="{\displaystyle Pf\in I^{\frac {n}{2}}({\mathfrak {so}}(n))}" loading="lazy"> erzeugt.

<div class="mw-heading mw-heading3"><h3 id="Beispiel_G=GL(n,ℂ)">Beispiel G=GL(n,ℂ)</h3></div>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy"> sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\frac {k}{2}}}">

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71e2f952e755283e4c05d93cf260c1a57ae90e88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.426ex; height:4.009ex;" alt="{\displaystyle C_{\frac {k}{2}}}" loading="lazy"> das komplex-wertige homogene Polynom vom Grad <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/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, dessen Wert auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,\ldots ,A)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc991cf6f116db66cdeccf44da9c20b2a8bcea78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.474ex; height:2.843ex;" alt="{\displaystyle (A,\ldots ,A)}" loading="lazy"> man als Koeffizienten vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy"> im Polynom

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)=\sum _{k}C_{k}(A,\ldots ,A)\lambda ^{n-k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d0493de830b0077f5d89ad949710c67c1f481f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.788ex; height:6.676ex;" alt="{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)=\sum _{k}C_{k}(A,\ldots ,A)\lambda ^{n-k}}" loading="lazy"></dd></dl>

erhält, für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \operatorname {Mat} (n,\mathbb {C} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (n,\mathbb {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52ddc24fd0ddd58197e977daea1d1eebf34af82f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.698ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (n,\mathbb {C} )}" loading="lazy">. Das Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a0887b56787ba96e79de2b9f5c6ff30aabad1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.751ex; height:2.509ex;" alt="{\displaystyle C_{k}}" loading="lazy"> heißt das <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/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-te Chern-Polynom. Die Chern- und Pontrjagin-Polynome hängen über die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{k}C_{k}(A,\ldots ,A)=P_{\frac {k}{2}}(A,\ldots ,A)}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{k}C_{k}(A,\ldots ,A)=P_{\frac {k}{2}}(A,\ldots ,A)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33cec6cb92926dd0e08a930a54c5bfa6226e55c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.944ex; height:4.509ex;" alt="{\displaystyle i^{k}C_{k}(A,\ldots ,A)=P_{\frac {k}{2}}(A,\ldots ,A)}" loading="lazy"> zusammen.

Die Algebra der komplex-wertigen invarianten Polynome wird von den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{k}\in I^{k}({\mathfrak {gl}}(n,\mathbb {C} ))}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{k}\in I^{k}({\mathfrak {gl}}(n,\mathbb {C} ))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961192d0e48a52d7b27d17db6f61837439810775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.445ex; height:3.176ex;" alt="{\displaystyle C_{k}\in I^{k}({\mathfrak {gl}}(n,\mathbb {C} ))}" loading="lazy"> erzeugt.

<div class="mw-heading mw-heading3"><h3 id="Beispiel_G=U(n)">Beispiel G=U(n)</h3></div>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathfrak {u}}(n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathfrak {u}}(n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/552e565020a9e96980bf625b61143dfb2018e8cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.99ex; height:2.843ex;" alt="{\displaystyle A\in {\mathfrak {u}}(n)}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=-A^{H}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a044b093d6a5001e75b7fe1f0d661a9d63dbaca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.084ex; height:2.843ex;" alt="{\displaystyle A=-A^{H}}" loading="lazy"> und damit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)={\overline {\det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)}},}">

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mover>

<mrow>

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)={\overline {\det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70e3541c37d038e361fc7f9959f23f47f317e144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.889ex; height:6.843ex;" alt="{\displaystyle \det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)={\overline {\det \left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)}},}" loading="lazy"> deshalb sind die Chern-Polynome auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {u}}(n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {u}}(n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a3f8b39878e3239985d192b7d46156b27cc79a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {u}}(n)}" loading="lazy"> reell-wertig.

Die Algebra der invarianten Polynome wird von den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{k}\in I^{k}({\mathfrak {u}}(n))}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{k}\in I^{k}({\mathfrak {u}}(n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80eea3d4afbd3c2c538af796fd43cf04ab28aa1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.112ex; height:3.176ex;" alt="{\displaystyle C_{k}\in I^{k}({\mathfrak {u}}(n))}" loading="lazy"> erzeugt.

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

<ul><li><a href="Shoshichi_Kobayashi" class="mw-redirect" title="Shoshichi Kobayashi">Shoshichi Kobayashi</a>, <a href="Katsumi_Nomizu" class="mw-redirect" title="Katsumi Nomizu">Katsumi Nomizu</a>: Foundations of differential geometry. Vol. I, II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969.</li>

<li>Johan L. Dupont: Curvature and characteristic classes. Lecture Notes in Mathematics, Vol. 640. Springer-Verlag, Berlin-New York, 1978. ISBN 3-540-08663-3</li></ul></div><div><div class="zim-footer">

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