Regular Representation - Finite Groups

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 G → K. 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 f : G → K and an element g ∈ G,

and