Group Representations

Abstract

We recall that a representation of a group G is a homomorphism of G to the group Aut L of linear transformations of some vector space L (see § 9); this notion is closely related to the idea of ‘coordinatisation’. The meaning of coordinatisation is to specify objects forming a homogeneous set X by assigning individually distinguishable quantities to them. Of course such a specification is in principle impossible: considering the inverse map would then make the objects of X themselves individually distinguishable. The resolution of this contradiction is that, in the process of coordinatisation, apart from the objects and the quantities, there is in fact always a third ingredient, the coordinate system (in one or other sense of the world), which is like a kind of physical measuring instrument. Only after fixing a coordinate system S can one assign to a given object x ∈ X a definite quantity, its ‘generalised coordinate’. But then the fundamental problem arises: how to distinguish the properties of the quantities that reflect properties of the objects themselves from those introduced by the choice of the coordinate system? This is the problem of invariance of various relations arising in theories of this kind. In spirit, it is entirely analogous to the problem of the observer in theoretical physics.