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In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex general linear group the theorem was first proved by who gave a case-by-case proof. soon afterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre. == Statement of the theorem == Let ''V'' be a finite-dimensional vector space over a field ''K'' and let ''G'' be a finite subgroup of the general linear group ''GL''(''V''). An element ''s'' of ''GL''(''V'') is called a pseudoreflection if it fixes a codimension 1 subspace of ''V'' and is not the identity transformation ''I'', or equivalently, if the kernel Ker (''s'' − ''I'') has codimension one in ''V''. Assume that the order of ''G'' is relatively prime to the characteristic of ''K'' (the so-called non-modular case). Then the following properties are equivalent:〔See, e.g.: Bourbaki, ''Lie'', chap. V, §5, nº5, theorem 4 for equivalence of (A), (B) and (C); page 26 of () for equivalence of (A) and (B′); pages 6–18 of () for equivalence of (C) and (C′) () for a proof of (B′)⇒(A).〕 * (A) The group ''G'' is generated by pseudoreflections. * (B) The algebra of invariants ''K''()''G'' is a (free) polynomial algebra. * (B′) The algebra of invariants ''K''()''G'' is a regular ring. * (C) The algebra ''K''() is a free module over ''K''()''G''. * (C′) The algebra ''K''() is a projective module over ''K''()''G''. In the case when the field ''K'' is the field C of complex numbers, the first condition is usually stated as "''G'' is a complex reflection group". Shephard and Todd derived a full classification of such groups. 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「Chevalley–Shephard–Todd theorem」の詳細全文を読む スポンサード リンク
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