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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 Young symmetrizer ： ウィキペディア英語版 Young symmetrizer In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space $V^$ obtained from the action of $S\_n$ on $V^$ by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young. ==Definition== Given a finite symmetric group ''S''''n'' and specific Young tableau λ corresponding to a numbered partition of ''n'', define two permutation subgroups $P\_\backslash lambda$ and $Q\_\backslash lambda$ of ''S''''n'' as follows: :$P\_\backslash lambda=\backslash$ and :$Q\_\backslash lambda=\backslash .$ Corresponding to these two subgroups, define two vectors in the group algebra $\backslash mathbbS\_n$ as :$a\_\backslash lambda=\backslash sum\_\; e\_g$ and :$b\_\backslash lambda=\backslash sum\_\; \backslash sgn(g)\; e\_g$ where $e\_g$ is the unit vector corresponding to ''g'', and $\backslash sgn(g)$ is the sign of the permutation. The product :$c\_\backslash lambda\; :=\; a\_\backslash lambda\; b\_\backslash lambda\; =\; \backslash sum\_\; \backslash sgn(h)\; e\_$ is the Young symmetrizer corresponding to the Young tableau λ. Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer. (If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.) 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「Young symmetrizer」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.