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In mathematics, a complex reflection group is a group acting on a finite-dimensional complex vector space, that is generated by complex reflections: non-trivial elements that fix a complex hyperplane in space pointwise. (Complex reflections are sometimes called pseudo reflections or unitary reflections or sometimes just reflections.) ==Classification== Any real reflection group becomes a complex reflection group if we extend the scalars from R to C. In particular all Coxeter groups or Weyl groups give examples of complex reflection groups. Any finite complex reflection group is a product of irreducible complex reflection groups, acting on the sum of the corresponding vector spaces. So it is sufficient to classify the irreducible complex reflection groups. The finite irreducible complex reflection groups were classified by . They found an infinite family ''G''(''m'',''p'',''n'') depending on 3 positive integer parameters (with ''p'' dividing ''m''), and 34 exceptional cases, that they numbered from 4 to 37, listed below. The group ''G''(''m'',''p'',''n''), of order ''m''''n''''n''!/''p'', is the semidirect product of the abelian group of order ''m''''n''/''p'' whose elements are (θ''a''1,θ''a''2, ...,θ''a''''n''), by the symmetric group ''S''''n'' acting by permutations of the coordinates, where θ is a primitive ''m''th root of unity and Σ''a''''i''≡ 0 mod ''p''; it is an index ''p'' subgroup of the generalized symmetric group Special cases of ''G''(''m'',''p'',''n''): *''G''(''1'',''1'',''n'') is the Coxeter group ''A''''n''−1 *''G''(''2'',''1'',''n'') is the Coxeter group ''B''''n'' = ''C''''n'' *''G''(''2'',''2'',''n'') is the Coxeter group ''D''''n'' *''G''(''m'',''p'',''1'') is a cyclic group of order ''m''/''p''. *''G''(''m'',''m'',''2'') is the Coxeter group ''I''''2''(''m'') (and the Weyl group ''G''2 when ''m'' = 6). *The group ''G''(''m'',''p'',''n'') acts irreducibly on C''n'' except in the cases ''m''=1, ''n''>1 (symmetric group) and ''G''(2,2,2) (Klein 4 group), when C''n'' splits as a sum of irreducible representations of dimensions 1 and ''n''−1. *The only cases when two groups ''G''(''m'',''p'',''n'') are isomorphic as complex reflection groups are that ''G''(''ma'',''pa'',1) is isomorphic to ''G''(''mb'',''pb'',1) for any positive integers ''a'',''b''. However there are other cases when two such groups are isomorphic as abstract groups. *The complex reflection group ''G''(2,2,3) is isomorphic as a complex reflection group to ''G''(1,1,4) restricted to a 3-dimensional space. *The complex reflection group ''G''(3,3,2) is isomorphic as a complex reflection group to ''G''(1,1,3) restricted to a 2-dimensional space. *The complex reflection group ''G''(2''p'',''p'',1) is isomorphic as a complex reflection group to ''G''(1,1,2) restricted to a 1-dimensional space. 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「Complex reflection group」の詳細全文を読む スポンサード リンク
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