1.3.2 Group representations
Groups often occur as “transformation groups”, meaning groups of elements
acting as transformations of some particular geometric object. In the example
mentioned above of the group of three dimensional rotations, such rotations are
linear transformations ofR^3. In general:
Definition(Group action on a set). An action of a groupGon a setMis
given by a map
(g,x)∈G×M→g·x∈M
that takes a pair(g,x)of a group elementg∈Gand an elementx∈Mto
another elementg·x∈Msuch that
g 1 ·(g 2 ·x) = (g 1 g 2 )·x (1.2)and
e·x=x
whereeis the identity element ofG
A good example to keep in mind is that of three dimensional spaceM=R^3
with the standard inner product. This comes with two different group actions
preserving the inner product
- An action of the groupG 1 =R^3 onR^3 by translations.
- An action of the groupG 2 =O(3) of three dimensional orthogonal trans-
formations ofR^3. These are the rotations about the origin (possibly
combined with a reflection). Note that in this case order matters: for
non-commutative groups likeO(3) one hasg 1 g 26 =g 2 g 1 for some group
elementsg 1 ,g 2.
A fundamental principle of modern mathematics is that the way to under-
stand a spaceM, given as some set of points, is to look atF(M), the set of
functions on this space. This “linearizes” the problem, since the function space
is a vector space, no matter what the geometrical structure of the original set
is. If the set has a finite number of elements, the function space will be a finite
dimensional vector space. In general though it will be infinite dimensional and
one will need to further specify the space of functions (e.g., continuous functions,
differentiable functions, functions with finite norm, etc.) under consideration.
Given a group action ofGonM, functions onMcome with an action ofG
by linear transformations, given by
(g·f)(x) =f(g−^1 ·x) (1.3)wherefis some function onM.