• As we already mentioned at the beginning of Section 10.1, groups always occur as groups of transformations of something, through their action on a set (usually endowed with some additional structure). Thus, there exists a rule which assigns to each element g of a group a transformation Lg of some set M. A study of group theory thus naturally incorporates181 besides knowledge of the groups themselves also the question of where and how a given group may act.

In this chapter we will systematically treat a particular, but very important, class of actions, which are called representations. From the perspective of general actions, to be discussed in more detail in Chapter 13, they are singled out by operating in linear spaces and, moreover, linearly. Such a distinguished position of just this class of actions within the scope of all actions is simply the reflection of the distinguished position of linear spaces within the scope of various mathematical structures. Representations may be found wherever symmetries and linearity meet in one place.

Basic concepts

• If a symmetry group is to act in a linear space V, it is natural to ask for the compatibility of the symmetry operations with the linear structure. This means that to each group element g we should assign a linear operator ρ(g), i.e. ρ(g) ∈ End V. Moreover, these maps should also “reproduce” the behavior of the abstract group G itself, i.e. to be homomorphisms from G to End V.