Finite groups Regular representation

for finite group g, left regular representation λ (over field k) linear representation on k-vector space v freely generated elements of g, i. e. can identified basis of v. given g ∈ g, λ(g) linear map determined action on basis left translation g, i.e.

λ
(
g
)
:
h
↦
g
h
,

 for all 

h
∈
g
.


{\displaystyle \lambda (g):h\mapsto gh,{\text{ }}h\in g.}

for right regular representation ρ, inversion must occur in order satisfy axioms of representation. specifically, given g ∈ g, ρ(g) linear map on v determined action on basis right translation g, i.e.

ρ
(
g
)
:
h
↦
h

g

−
1


,

 for all 

h
∈
g
.
 


{\displaystyle \rho (g):h\mapsto hg^{-1},{\text{ }}h\in g.\ }

alternatively, these representations can defined on k-vector space w of functions g → k. in form regular representation generalized topological groups such lie groups.

the specific definition in terms of w follows. given function f : g → k , element g ∈ g,

(
λ
(
g
)
f
)
(
x
)
=
f
(

g

−
1


x
)


{\displaystyle (\lambda (g)f)(x)=f(g^{-1}x)}

and

(
ρ
(
g
)
f
)
(
x
)
=
f
(
x
g
)
.


{\displaystyle (\rho (g)f)(x)=f(xg).}