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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 Longest element of a Coxeter group ： ウィキペディア英語版 Longest element of a Coxeter group In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by ''w''0. See and . == Properties == * A Coxeter group has a longest element if and only if it is finite; "only if" is because the size of the group is bounded by the number of words of length less than or equal to the maximum. * The longest element of a Coxeter group is the unique maximal element with respect to the Bruhat order. * The longest element is an involution (has order 2: $w\_0^\; =\; w\_0$), by uniqueness of maximal length (the inverse of an element has the same length as the element). * For any $w\; \backslash in\; W,$ the length satisfies $\backslash ell(w\_0w)\; =\; \backslash ell(w\_0)\; -\; \backslash ell(w).$〔 * A reduced expression for the longest element is not in general unique. * In a reduced expression for the longest element, every simple reflection must occur at least once.〔 * If the Coxeter group is a finite Weyl group then the length of ''w''0 is the number of the positive roots. * The open cell ''Bw''0''B'' in the Bruhat decomposition of a semisimple algebraic group ''G'' is dense in Zariski topology; topologically, it is the top dimensional cell of the decomposition, and represents the fundamental class. * The longest element is the central element –1 except for $A\_n$ ($n\; \backslash geq\; 2$), $D\_n$ for ''n'' odd, $E\_6,$ and $I\_2(p)$ for ''p'' odd, when it is –1 multiplied by the order 2 automorphism of the Coxeter diagram. 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「Longest element of a Coxeter group」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.