Editing Regular representation (section)

==Finite groups==
{{See also|Representation theory of finite groups#Left- and right-regular representation}}
For a [[finite group]] ''G'', the left regular representation λ (over a [[field (mathematics)|field]] ''K'') is a linear representation on the [[vector space|''K''-vector space]] ''V'' freely generated by the elements of ''G'', i.e. elements of ''G'' can be identified with a [[basis (linear algebra)|basis]] of ''V''. Given ''g''&nbsp;∈&nbsp;''G'', λ<sub>''g''</sub> is the linear map determined by its action on the basis by left translation by ''g'', i.e.

:<math>\lambda_{g}:h\mapsto gh,\text{ for all }h\in G.</math>

For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given ''g''&nbsp;∈&nbsp;''G'', ρ<sub>''g''</sub> is the linear map on ''V'' determined by its action on the basis by right translation by ''g''<sup>&minus;1</sup>, i.e.

:<math>\rho_{g}:h\mapsto hg^{-1},\text{ for all }h\in G.\ </math>

Alternatively, these representations can be defined on the ''K''-vector space ''W'' of all functions {{nowrap|''G'' → ''K''}}. It is in this form that the regular representation is generalized to [[topological group]]s such as [[Lie group]]s.

The specific definition in terms of ''W'' is as follows. Given a function {{nowrap|''f'' : ''G'' → ''K''}} and an element ''g''&nbsp;∈&nbsp;''G'',
:<math>(\lambda_{g}f)(x)=f(\lambda_{g}^{-1}(x))=f({g}^{-1}x)</math>
and
:<math>(\rho_{g}f)(x)=f(\rho_{g}^{-1}(x))=f(xg).</math>