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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 Finite Coxeter group ： ウィキペディア英語版 Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard references include and . ==Definition== Formally, a Coxeter group can be defined as a group with the presentation :$\backslash left\backslash langle\; r\_1,r\_2,\backslash ldots,r\_n\; \backslash mid\; (r\_ir\_j)^=1$ and $m\_\backslash geq\; 2$ for $i\backslash neq\; j$. The condition $m\_=\; \backslash infty$ means no relation of the form $(r\_i\; r\_j)^m$ should be imposed. The pair ''(W,S)'' where ''W'' is a Coxeter group with generators ''S''= is called Coxeter system. Note that in general ''S'' is ''not'' uniquely determined by ''W''. For example, the Coxeter groups of type ''BC''''3'' and ''A''''1''x''A''''3'' are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation). A number of conclusions can be drawn immediately from the above definition. * The relation ''m''''i i'' = 1 means that (''r''''i''''r''''i'' )1 = (''r''''i'' )2 = 1 for all ''i'' ; the generators are involutions. * If ''m''''i j'' = 2, then the generators ''r''''i'' and ''r''''j'' commute. This follows by observing that ::''xx'' = ''yy'' = 1, : together with :: ''xyxy'' = 1 : implies that :: ''xy'' = ''x''(''xyxy'')''y'' = (''xx'')''yx''(''yy'') = ''yx''. :Alternatively, since the generators are involutions, $r\_i\; =\; r\_i^$, so $(r\_ir\_j)^2=r\_ir\_jr\_ir\_j=r\_ir\_jr\_i^r\_j^$, and thus is equal to the commutator. * In order to avoid redundancy among the relations, it is necessary to assume that ''mi j'' = ''mj i''. This follows by observing that ::''yy'' = 1, : together with :: (''xy'')''m'' = 1 : implies that :: (''yx'')''m'' = (''yx'')''m''''yy'' = ''y''(''xy'')''m''''y'' = ''yy'' = 1. :Alternatively, $(xy)^k$ and $(yx)^k$ are conjugate elements, as $y(xy)^k\; y^\; =\; (yx)^k\; yy^=(yx)^k$. 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 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.