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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Weyl character formula ： ウィキペディア英語版 Weyl character formula 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 . By definition, the character of a representation ''r'' of ''G'' is the trace of ''r''(''g''), as a function of a group element ''g'' in ''G''. 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 χ of ''r'' is a good substitute for ''r'' itself, and can have algorithmic content. Weyl's formula is a closed formula for the χ, in terms of other objects constructed from ''G'' and its Lie algebra. The representations in question here are complex, and so without loss of generality are unitary representations; ''irreducible'' therefore means the same as ''indecomposable'', i.e. not a direct sum of two subrepresentations. ==Statement of Weyl character formula== The character of an irreducible representation $V$ of a complex semisimple Lie algebra $\backslash mathfrak$ is given by :$\backslash operatorname(V)\; =\; \backslash frac\}\}(1-e^)\}$ where *$W$ is the Weyl group; *$\backslash Delta^$ is the subset of the positive roots of the root system $\backslash Delta$; *$\backslash rho$ is the half sum of the positive roots; *$\backslash lambda$ is the highest weight of the irreducible representation $V$; *$\backslash varepsilon(w)$ is the determinant of the action of $w$ on the Cartan subalgebra $\backslash mathfrak\; \backslash subset\; \backslash mathfrak$. This is equal to $(-1)^$, where $\backslash ell(w)$ is the length of the Weyl group element, defined to be the minimal number of reflections with respect to simple roots such that $w$ equals the product of those reflections. The character of an irreducible representation $V$ of a compact connected Lie group $G$ is given by :$\backslash operatorname(V)\; =\; \backslash frac\})\}$ where $\backslash xi\_$ is the character on $T$ with differential $\backslash alpha$ on the Lie algebra $\backslash mathfrak\_$ of the maximal Torus $T$. If $\backslash rho$ is the differential of a character of $T$, e.g. if $G$ is simply connected, this can be reformulated as :$\backslash operatorname(V)\; =\; \backslash frac\}\}(1-\backslash xi\_)\}\; =\; \backslash frac\}\}$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Weyl character formula」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.