The order in which elements of the group act may matter, so the inverse is
needed to get the group action property 1.2, since
g 1 ·(g 2 ·f)(x) = (g 2 ·f)(g 1 −^1 ·x)
=f(g 2 −^1 ·(g− 11 ·x))
=f((g− 21 g− 11 )·x)
=f((g 1 g 2 )−^1 ·x)
= (g 1 g 2 )·f(x)This calculation would not work out properly for non-commutativeGif one
defined (g·f)(x) =f(g·x).
One can abstract from this situation and define as follows a representation
as an action of a group by linear transformations on a vector space:
Definition(Representation).A representation(π,V)of a groupGis a homo-
morphism
π:g∈G→π(g)∈GL(V)
whereGL(V)is the group of invertible linear mapsV →V, withV a vector
space.
Saying the mapπis a homomorphism means
π(g 1 )π(g 2 ) =π(g 1 g 2 )for allg 1 ,g 2 ∈G, i.e., that it satisfies the property needed to get a group action.
We will mostly be interested in the case of complex representations, whereVis a
complex vector space, so one should assume from now on that a representation is
complex unless otherwise specified (there will be cases where the representations
are real).
WhenVis finite dimensional and a basis ofVhas been chosen, then linear
maps and matrices can be identified (see the review of linear algebra in chapter
4). Such an identification provides an isomorphism
GL(V)'GL(n,C)of the group of invertible linear maps ofVwithGL(n,C), the group of invertible
nbyncomplex matrices. We will begin by studying representations that are
finite dimensional and will try to make rigorous statements. Later on we will
get to representations on function spaces, which are infinite dimensional, and
will then often neglect rigor and analytical difficulties. Note that only in the
case ofM a finite set of points will we get an action by finite dimensional
matrices this way, since thenF(M) will be a finite dimensional vector space
(C# of points in M).
A good example to consider to understand this construction in the finite
dimensional case is the following: