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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 Schur–Weyl duality ： ウィキペディア英語版 Schur–Weyl duality Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. It is named after two pioneers of representation theory of Lie groups, Issai Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups as a way of classifying representations of unitary and general linear groups. == Description == Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space : $\backslash mathbb^n\backslash otimes\backslash mathbb^n\backslash otimes\backslash cdots\backslash otimes\backslash mathbb^n$ with ''k'' factors. The symmetric group ''S''''k'' on ''k'' letters acts on this space (on the left) by permuting the factors, : $\backslash sigma(v\_1\backslash otimes\; v\_2\backslash otimes\backslash cdots\backslash otimes\; v\_k)\; =\; v\_\backslash otimes\; v\_\backslash otimes\backslash cdots\backslash otimes\; v\_.$ The general linear group ''GL''''n'' of invertible ''n''×''n'' matrices acts on it by the simultaneous matrix multiplication, : $g(v\_1\backslash otimes\; v\_2\backslash otimes\backslash cdots\backslash otimes\; v\_k)\; =\; gv\_1\backslash otimes\; gv\_2\backslash otimes\backslash cdots\backslash otimes\; gv\_k,\; \backslash quad\; g\backslash in\; GL\_n.$ These two actions commute, and in its concrete form, the Schur–Weyl duality asserts that under the joint action of the groups ''S''''k'' and ''GL''''n'', the tensor space decomposes into a direct sum of tensor products of irreducible modules for these two groups that determine each other, : $\backslash mathbb^n\backslash otimes\backslash mathbb^n\backslash otimes\backslash cdots\backslash otimes\backslash mathbb^n\; =\; \backslash sum\_D\; \backslash pi\_k^D\backslash otimes\backslash rho\_n^D.$ The summands are indexed by the Young diagrams ''D'' with ''k'' boxes and at most ''n'' rows, and representations $\backslash pi\_k^D$ of ''S''''k'' with different ''D'' are mutually non-isomorphic, and the same is true for representations $\backslash rho\_n^D$ of ''GL''''n''. The abstract form of the Schur–Weyl duality asserts that two algebras of operators on the tensor space generated by the actions of ''GL''''n'' and ''S''''k'' are the full mutual centralizers in the algebra of the endomorphisms $\backslash mathrm\_\backslash mathbb(\backslash mathbb^n\backslash otimes\backslash mathbb^n\backslash otimes\backslash cdots\backslash otimes\backslash mathbb^n).$ 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「Schur–Weyl duality」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.