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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 Jucys–Murphy element ： ウィキペディア英語版 Jucys–Murphy element In mathematics, the Jucys–Murphy elements in the group algebra $\backslash mathbb\; ()$ of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula: :$X\_1=0,\; ~~~\; X\_k=\; (1\; k)+\; (2\; k)+\backslash cdots+(k-1\backslash \; k),\; ~~~\; k=2,\backslash dots,n.$ They play an important role in the representation theory of the symmetric group. ==Properties== They generate a commutative subalgebra of $\backslash mathbb\; (S\_n)$. Moreover, ''X''''n'' commutes with all elements of $\backslash mathbb\; ()$. The vectors of the Young basis are eigenvectors for the action of ''X''''n''. For any standard Young tableau ''U'' we have: :$X\_k\; v\_U\; =c\_k(U)\; v\_U,\; ~~~\; k=1,\backslash dots,n,$ where ''c''''k''(''U'') is the ''content'' ''b'' − ''a'' of the cell (''a'', ''b'') occupied by ''k'' in the standard Young tableau ''U''. Theorem (Jucys): The center $Z(\backslash mathbb\; ())$ of the group algebra $\backslash mathbb\; ()$ of the symmetric group is generated by the symmetric polynomials in the elements ''Xk''. Theorem (Jucys): Let ''t'' be a formal variable commuting with everything, then the following identity for polynomials in variable ''t'' with values in the group algebra $\backslash mathbb\; ()$ holds true: :$(t+X\_1)\; (t+X\_2)\; \backslash cdots\; (t+X\_n)=\; \backslash sum\_\; \backslash sigma\; t^.$ Theorem (Okounkov–Vershik): The subalgebra of $\backslash mathbb\; ()$ generated by the centers : $Z(\backslash mathbb\; (S\_1)),\; Z(\backslash mathbb\; (S\_2)),\; \backslash ldots,\; Z(\backslash mathbb\; (S\_)),\; Z(\backslash mathbb\; ())$ is exactly the subalgebra generated by the Jucys–Murphy elements ''Xk''. 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「Jucys–Murphy element」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.