翻訳と辞書 Words near each other ・ Equivalent temperature ・ Equivalent territory ・ Equivalent variation ・ Equivalent VIII ・ Equivalent weight ・ Equivalent width ・ Equivalents ・ Equivalents of duke in other European languages ・ Equivalisation ・ Equivariant bundle ・ Equivariant cohomology ・ Equivariant differential form ・ Equivariant index theorem ・ Equivariant K-theory ・ Equivariant L-function ・ Equivariant map ・ Equivariant sheaf ・ Equivariant stable homotopy theory ・ Equivariant topology ・ Equiveillance ・ Equivocada ・ Equivocation ・ Equivocation (disambiguation) ・ Equivocation (magic) ・ Equivocation (play) ・ Equix ・ Equmenia ・ Equo ・ Equol ・ Equord Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Equivariant map ： ウィキペディア英語版 Equivariant map In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let ''G'' be a group and let ''X'' and ''Y'' be two associated ''G''-sets. A function ''f'' : ''X'' → ''Y'' is said to be equivariant if :''f''(''g''·''x'') = ''g''·''f''(''x'') for all ''g'' ∈ ''G'' and all ''x'' in ''X''. Note that if one or both of the actions are right actions the equivariance condition must be suitably modified: :''f''(''x''·''g'') = ''f''(''x'')·''g'' ; (right-right) :''f''(''x''·''g'') = ''g''−1·''f''(''x'') ; (right-left) :''f''(''g''·''x'') = ''f''(''x'')·''g''−1 ; (left-right) Equivariant maps are homomorphisms in the category of ''G''-sets (for a fixed ''G''). Hence they are also known as ''G''-maps or ''G''-homomorphisms. Isomorphisms of ''G''-sets are simply bijective equivariant maps. The equivariance condition can also be understood as the following commutative diagram. Note that $g\backslash cdot$ denotes the map that takes an element $z$ and returns $g\backslash cdot\; z$. ==Intertwiners== A completely analogous definition holds for the case of linear representations of ''G''. Specifically, if ''X'' and ''Y'' are the representation spaces of two linear representations of ''G'' then a linear map ''f'' : ''X'' → ''Y'' is called an intertwiner of the representations if it commutes with the action of ''G''. Thus an intertwiner is an equivariant map in the special case of two linear representations/actions. Alternatively, an intertwiner for representations of ''G'' over a field ''K'' is the same thing as a module homomorphism of ''K''()-modules, where ''K''() is the group ring of ''G''. Under some conditions, if ''X'' and ''Y'' are both irreducible representations, then an intertwiner (other than the zero map) only exists if the two representations are equivalent (that is, are isomorphic as modules). That intertwiner is then unique up to a multiplicative factor (a non-zero scalar from ''K''). These properties hold when the image of ''K''() is a simple algebra, with centre K (by what is called Schur's Lemma: see simple module). As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same. 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「Equivariant map」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.