Group Representations

Abstract

Suppose we associate with each element of a group G a linear transformation T _g of a vector space E, in such a way that to the product of elements of G is associated the composition of the corresponding transformations:

$${T_{gh}} = {T_g}{T_h},forg,h \in G.$$

(39.1)

We then say that the correspondence g ↦ T _g , also denoted T, is a linear representation of G. In other words, T is a homomorphism from G into the group GL(E) of linear transformations of E. We also say that E is the representation space of G (under the representation T).