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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 Coxeter notation ： ウィキペディア英語版 Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. == Reflectional groups== For Coxeter groups defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors. The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the ''A''''n'' group is represented by (), to imply ''n'' nodes connected by ''n-1'' order-3 branches. Example ''A''2 = () = () = . Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like (), starting with () = as D4. Coxeter allowed for zeros as special cases to fit the ''A''''n'' family, like ''A''3 = () = () = () = (), like = = . Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, like () = for the triangle group (p q r). If the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like () = , representing Coxeter diagram or . can be represented as () or . More complicated looping diagrams can also be expressed with care. The paracompact Coxeter group can be represented by Coxeter notation (), with nested/overlapping parentheses showing two adjacent () loops, and is also represented more compactly as ], representing the rhombic symmetry of the Coxeter diagram. The paracompact complete graph diagram or , is represented as with () as the symmetry of its regular tetrahedron coxeter diagram. The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter diagram = ''A''2×''A''2 = ''A''22 can be represented by ()×() = ()2 = (). }_1 || () || |- | $\}\_2$ || () || |- | $\}\_3$ || || |- | $\}\_3$ || () || |- | $\}\_4$ || () || |- | $\}\_4$ || (31,1,1,1 ) || |- | $\}\_n$ || || ... or ... |- | $\}\_n$ || () || ... |- | $\}\_6$ || () || |- | $\}\_8$ || () || |} | }_3||() || |- | $\}\_3$||() || |- | $\}\_3$ || () ||   |- | $\}\_3$ || () || |- | $\}\_4$ || () || |- | $\}\_4$ || () || |- | $\}\_4$ || () || |} |} For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's 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「Coxeter notation」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.