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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 Littlewood–Richardson coefficient ： ウィキペディア英語版 Littlewood–Richardson rule In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of irreducible representations of general linear groups (or related groups like the special linear and special unitary groups), or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say $\backslash lambda,\backslash mu,\backslash nu$, of which $\backslash lambda$ and $\backslash mu$ describe the Schur functions being multiplied, and $\backslash nu$ gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients $c\_^\backslash nu$ such that :$s\_\backslash lambda\; s\_\backslash mu=\backslash sum\_\backslash nu\; c\_^\backslash nu\; s\_\backslash nu.$ The Littlewood–Richardson rule states that $c\_^\backslash nu$ is equal to the number of Littlewood–Richardson tableaux of skew shape $\backslash nu/\backslash lambda$ and of weight $\backslash mu$. == History == The Littlewood–Richardson rule was first stated by but though they claimed it as a theorem they only proved it in some fairly simple special cases. claimed to complete their proof, but his argument had gaps, though it was so obscurely written that these gaps were not noticed for some time, and his argument is reproduced in the book . Some of the gaps were later filled by . The first rigorous proofs of the rule were given four decades after it was found, by and , after the necessary combinatorial theory was developed by , , and in their work on the Robinson–Schensted correspondence. There are now several short proofs of the rule, such as , and using Bender-Knuth involutions. used the Littelmann path model to generalize the Littlewood–Richardson rule to other semisimple Lie groups. The Littlewood–Richardson rule is notorious for the number of errors that appeared prior to its complete, published proof. Several published attempts to prove it are incomplete, and it is particularly difficult to avoid errors when doing hand calculations with it: even the original example in contains an error. 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「Littlewood–Richardson rule」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.