| dbo:abstract |
In der Mathematik ist die Weylsche Charakterformel oder Charakterformel von Weyl eine Formel zur Berechnung des Charakters einer Darstellung aus ihrem höchsten Gewicht. Sie wurde 1926 von Hermann Weyl bewiesen und folgt auch aus dem Atiyah-Bott-Fixpunktsatz. (de) In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula. By definition, the character of a representation of G is the trace of , as a function of a group element . The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. Knowledge of the character of gives a lot of information about itself. Weyl's formula is a closed formula for the character , in terms of other objects constructed from G and its Lie algebra. (en) 리 군 표현론에서 바일 지표 공식(Weyl指標公式, 영어: Weyl character formula)은 주어진 복소수 기약 표현의 지표를 제시하는 공식이다. (ko) 数学において，表現論におけるワイルの指標公式（英: Weyl character formula）はコンパクトリー群の既約表現の指標をのことばで記述する．Hermann Weyl によって証明された． 定義により，G の表現 r の指標は群 G の元 g の関数としての r(g) のトレースである．この場合既約表現はすべて有限次元である（これはの一部である）．よってトレースの概念は線型代数学の通常のものである．r の指標 ξ を知ることは r 自身の良い代替であり，アルゴリズム的内容を持ち得る．ワイルの公式は G から構成される他の対象と G のリー環のことばで ξ をで表す．ここで問題の表現は複素でありしたがって一般性を失うことなくユニタリ表現である；したがって既約は直既約，つまり2つの部分表現の直和でないことと同じ意味である． (ja) 外尔特徵標公式（Weyl's character formula) 描述緊李羣不可約表示的特徵標。其名來自證明者赫尔曼·外尔。 定義：羣G的表示r的特徵標為一函數 ，，其中Tr 為線性算子之迹。 （由彼得-外尔定理 可知緊李羣的任何不可約表示都是有限維的；故迹之定義為線性代數中之定義。） 特徵標 χ 記住了表示 r 本身的重要訊息。 外尔特徵標公式用羣G的其他資料來表達 χ 。 本文考慮複表示，不失一般亦設其為，因而「不可約」亦等價於「不可分解」（即非二子表示之直和）。 (zh) |
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Duncan J. Melville (en) |
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Hermann Weyl (en) |
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Hermann (en) |
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W/w130070 (en) |
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Weyl (en) |
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Weyl–Kac character formula (en) |
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In der Mathematik ist die Weylsche Charakterformel oder Charakterformel von Weyl eine Formel zur Berechnung des Charakters einer Darstellung aus ihrem höchsten Gewicht. Sie wurde 1926 von Hermann Weyl bewiesen und folgt auch aus dem Atiyah-Bott-Fixpunktsatz. (de) 리 군 표현론에서 바일 지표 공식(Weyl指標公式, 영어: Weyl character formula)은 주어진 복소수 기약 표현의 지표를 제시하는 공식이다. (ko) 数学において，表現論におけるワイルの指標公式（英: Weyl character formula）はコンパクトリー群の既約表現の指標をのことばで記述する．Hermann Weyl によって証明された． 定義により，G の表現 r の指標は群 G の元 g の関数としての r(g) のトレースである．この場合既約表現はすべて有限次元である（これはの一部である）．よってトレースの概念は線型代数学の通常のものである．r の指標 ξ を知ることは r 自身の良い代替であり，アルゴリズム的内容を持ち得る．ワイルの公式は G から構成される他の対象と G のリー環のことばで ξ をで表す．ここで問題の表現は複素でありしたがって一般性を失うことなくユニタリ表現である；したがって既約は直既約，つまり2つの部分表現の直和でないことと同じ意味である． (ja) 外尔特徵標公式（Weyl's character formula) 描述緊李羣不可約表示的特徵標。其名來自證明者赫尔曼·外尔。 定義：羣G的表示r的特徵標為一函數 ，，其中Tr 為線性算子之迹。 （由彼得-外尔定理 可知緊李羣的任何不可約表示都是有限維的；故迹之定義為線性代數中之定義。） 特徵標 χ 記住了表示 r 本身的重要訊息。 外尔特徵標公式用羣G的其他資料來表達 χ 。 本文考慮複表示，不失一般亦設其為，因而「不可約」亦等價於「不可分解」（即非二子表示之直和）。 (zh) In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula. (en) |
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Weylsche Charakterformel (de) 바일 지표 공식 (ko) ワイルの指標公式 (ja) Weyl character formula (en) 外尔特征标公式 (zh) |
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