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Representation theory of finite groups Darstellungstheorie endlicher Gruppen 有限群表示論 Θεωρία αναπαραστάσεων πεπερασμένων ομάδων Théorie des représentations d'un groupe fini
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The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on . En mathématiques et plus précisément en théorie des groupes, la théorie des représentations d'un groupe fini traite des représentations d'un groupe G dans le cas particulier où G est un groupe fini. Cet article traite de l'aspect mathématique et, de même que l'article de synthèse « Représentations d'un groupe fini », n'aborde que les représentations linéaires de G (par opposition aux représentations projectives ou (en)). Ce sont les actions linéaires de G sur un espace vectoriel V de dimension finie, ou encore, les morphismes de G vers le groupe général linéaire GL(V) des automorphismes de V. 在數學裡,表示理論是以線性變換的群來分析一般抽象群的一種技術。相關的介紹請見群表示,此條目則討論含有有限個元素的群的表示理論。 表示論也在諸多領域上有應用,例如說:量子化學或是量子物理等等。除此之外,有限群表示論也常應用在代數上去檢驗群的結構,甚至在其他數學領域上,例如調和分析或是數論上,都是有應用的。 Die Darstellungstheorie endlicher Gruppen ist ein Teilgebiet der Mathematik, in dem man untersucht, wie Gruppen auf gegebenen Strukturen operieren. Man betrachtet vor allem die Operationen von Gruppen auf Vektorräumen (lineare Darstellungen). Allerdings werden auch die Operationen von Gruppen auf anderen Gruppen oder auf Mengen (Permutationsdarstellung) betrachtet. Weite Teile der Darstellungstheorie endlicher Gruppen lassen sich zur Darstellungstheorie kompakter Gruppen verallgemeinern. Στα μαθηματικά, η θεωρία αναπαραστάσεων είναι μια τεχνική για την ανάλυση των αφηρημένων ομάδων όσον αφορά τις ομάδες των γραμμικών μετασχηματισμών. Αυτό το άρθρο ασχολείται με την θεωρία αναπαραστάσεων των ομάδων που έχουν ένα πεπερασμένο αριθμό στοιχείων.
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Στα μαθηματικά, η θεωρία αναπαραστάσεων είναι μια τεχνική για την ανάλυση των αφηρημένων ομάδων όσον αφορά τις ομάδες των γραμμικών μετασχηματισμών. Αυτό το άρθρο ασχολείται με την θεωρία αναπαραστάσεων των ομάδων που έχουν ένα πεπερασμένο αριθμό στοιχείων. The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on . Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to gain new results about automorphic forms. En mathématiques et plus précisément en théorie des groupes, la théorie des représentations d'un groupe fini traite des représentations d'un groupe G dans le cas particulier où G est un groupe fini. Cet article traite de l'aspect mathématique et, de même que l'article de synthèse « Représentations d'un groupe fini », n'aborde que les représentations linéaires de G (par opposition aux représentations projectives ou (en)). Ce sont les actions linéaires de G sur un espace vectoriel V de dimension finie, ou encore, les morphismes de G vers le groupe général linéaire GL(V) des automorphismes de V. Die Darstellungstheorie endlicher Gruppen ist ein Teilgebiet der Mathematik, in dem man untersucht, wie Gruppen auf gegebenen Strukturen operieren. Man betrachtet vor allem die Operationen von Gruppen auf Vektorräumen (lineare Darstellungen). Allerdings werden auch die Operationen von Gruppen auf anderen Gruppen oder auf Mengen (Permutationsdarstellung) betrachtet. In diesem Artikel werden bis auf gekennzeichnete Ausnahmen nur endliche Gruppen betrachtet. Wir beschränken uns außerdem bei den Darstellungsräumen auf Vektorräume über Grundkörpern der Charakteristik Die Theorie der algebraisch abgeschlossenen Körper der Charakteristik ist abgeschlossen, d. h., wenn eine Theorie für einen algebraisch abgeschlossenen Körper der Charakteristik gilt, so gilt sie auch für alle anderen. Damit können wir im Folgenden ohne Einschränkung Vektorräume über betrachten. Die Darstellungstheorie findet in vielen Bereichen der Mathematik sowie in der Quantenchemie und der Physik Anwendung.Man verwendet die Darstellungstheorie unter anderem in der Algebra, um die Struktur von Gruppen zu untersuchen, in der harmonischen Analyse und in der Zahlentheorie.Beispielsweise wird im modernen Ansatz zum besseren Verständnis automorpher Formen die Darstellungstheorie verwendet. Weite Teile der Darstellungstheorie endlicher Gruppen lassen sich zur Darstellungstheorie kompakter Gruppen verallgemeinern. 在數學裡,表示理論是以線性變換的群來分析一般抽象群的一種技術。相關的介紹請見群表示,此條目則討論含有有限個元素的群的表示理論。 表示論也在諸多領域上有應用,例如說:量子化學或是量子物理等等。除此之外,有限群表示論也常應用在代數上去檢驗群的結構,甚至在其他數學領域上,例如調和分析或是數論上,都是有應用的。
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