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
Read more about this topic: Regular Representation
Famous quotes containing the words finite and/or groups:
“Are not all finite beings better pleased with motions relative than absolute?”
—Henry David Thoreau (1817–1862)
“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists.”
—Thomas S. Kuhn (b. 1922)