Basic Notions of Algebra pp 160-177 | Cite as

# 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.

## Keywords

Abelian Group Irreducible Representation Finite Group Conjugacy Class Character Group## Preview

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