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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 Weingarten function ： ウィキペディア英語版 Weingarten function In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by who found their asymptotic behavior, and named by , who evaluated them explicitly for the unitary group. ==Unitary groups== Weingarten functions are used for evaluating integrals over the unitary group ''U''''d'' of products of matrix coefficients of the form :$\backslash int\_\; U\_\backslash cdots\; U\_U^$ *_\cdots U^ *_dU. (Here $U^$ * denotes the conjugate transpose of $U$, alternatively denoted as $U^\backslash dagger$.) This integral is equal to :$\backslash sum\_\backslash delta\_\}\; \backslash delta\_\}Wg(d,\backslash sigma\backslash tau^)$ where ''Wg'' is the Weingarten function, given by :$$ Wg(d,\sigma) = \frac\sum_\frac where the sum is over all partitions λ of ''q'' . Here χλ is the character if ''S''''q'' corresponding to the partition λ and ''s'' is the Schur polynomial of λ, so that ''s''λ''d''(1) is the dimension of the representation of ''U''''d'' corresponding to λ. The Weingarten functions are rational functions in ''d''. They can have poles for small values of ''d'', which cancel out in the formula above. There is an alternative inequivalent definition of Weingarten functions, where one only sums over partitions with at most ''d'' parts. This is no longer a rational function of ''d'', but is finite for all positive integers ''d''. The two sorts of Weingarten functions coincide for ''d'' larger than ''q'', and either can be used in the formula for the integral. 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「Weingarten function」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.