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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 Kronecker coefficient ： ウィキペディア英語版 Kronecker coefficient In mathematics, Kronecker coefficients ''g''λμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. More explicitly, given a partition λ of ''n'', write ''V''λ for the Specht module associated to λ. Then the Kronecker coefficients ''g''λμν are given by the rule :$V\_\backslash mu\; \backslash otimes\; V\_\backslash nu\; =\; \backslash bigoplus\_\backslash lambda\; g\_^\; V\_\backslash lambda.$ One can interpret this on the level of symmetric functions, giving a formula for the Kronecker product of two Schur polynomials: :$s\_\backslash mu\; \backslash star\; s\_\backslash nu\; =\; \backslash sum\_\; g\_^\; s\_\backslash lambda.$ This is to be compared with Littlewood–Richardson coefficients, where one instead considers the induced representation :$\backslash uparrow\_\}^^\; V\_\backslash lambda,$ and the corresponding operation of symmetric functions is the usual product. Also note that the Littlewood–Richardson coefficients are the analogue of the Kronecker coefficients for representations of GL''n'', i.e. if we write ''W''λ for the irreducible representation corresponding to λ (where λ has at most ''n'' parts), one gets that :$W\_\backslash mu\; \backslash otimes\; W\_\backslash nu\; =\; \backslash bigoplus\_\backslash lambda\; c\_^\; W\_\backslash lambda.$ showed that Kronecker coefficients are hard to compute. A big unsolved problem in representation theory and combinatorics is to give a combinatorial description of the Kronecker coefficients. The Kronecker coefficients may also be computed as :$$ g(\lambda,\mu, \nu) = \frac \sum_ \chi^(\sigma)\chi^(\sigma)\chi^(\sigma) and appear in the generalized Cauchy identity :$$ \sum_ g(\lambda,\mu ,\nu) s_\lambda(x)s_\mu(y)s_\nu(z) = \prod_ \frac. ==See also== *Littlewood–Richardson coefficient 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「Kronecker coefficient」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.