Representations of Groups

Abstract

Let G be a group and V be a vector space over a field F. A representation of G in V is V with a group action G × V →V such that for all g in G the assignment v ↦ g · v is an F-linear transformation on V. Let Ψ: G ↦ F* be a homomorphism where F* is the multiplicative group F’ - {0}. We have a one-dimensional representation F _Ψ which is F as a vector space and g · λ = Ψ(g) λ.