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In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. ==Definitions== Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number ''h'' of an irreducible root system. A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. *The Coxeter number is the number of roots divided by the rank. The number of reflections in the Coxeter group is half the number of roots. *The Coxeter number is the order of any Coxeter element;. *If the highest root is ∑''m''iα''i'' for simple roots α''i'', then the Coxeter number is 1 + ∑''m''i *The Coxeter number is the dimension of the corresponding Lie algebra is ''n''(''h'' + 1), where ''n'' is the rank and ''h'' is the Coxeter number. *The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials. *The Coxeter number is given by the following table: The invariants of the Coxeter group acting on polynomials form a polynomial algebra whose generators are the fundamental invariants; their degrees are given in the table above. Notice that if ''m'' is a degree of a fundamental invariant then so is ''h'' + 2 − ''m''. The eigenvalues of a Coxeter element are the numbers ''e''2π''i''(''m'' − 1)/''h'' as ''m'' runs through the degrees of the fundamental invariants. Since this starts with ''m'' = 2, these include the primitive ''h''th root of unity, ''ζh'' = ''e''2π''i''/''h'', which is important in the Coxeter plane, below. 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 element」の詳細全文を読む スポンサード リンク
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