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In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation. ==Finite groups== For a finite group ''G'', the left regular representation λ (over a field ''K'') is a linear representation on the ''K''-vector space ''V'' freely generated by the elements of ''G'', i. e. they can be identified with a basis of ''V''. Given ''g'' ∈ ''G'', λ(''g'') is the linear map determined by its action on the basis by left translation by ''g'', i.e. : For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given ''g'' ∈ ''G'', ρ(''g'') is the linear map on ''V'' determined by its action on the basis by right translation by ''g''−1, i.e. : Alternatively, these representations can be defined on the ''K''-vector space ''W'' of all functions . It is in this form that the regular representation is generalized to topological groups such as Lie groups. The specific definition in terms of ''W'' is as follows. Given a function and an element ''g'' ∈ ''G'', : and : 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「Regular representation」の詳細全文を読む スポンサード リンク
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