Chapter 2 Representation theory
ΒΆWhat is representation theory? So far we have studied groups as abstract objects, defined by a set and some operation on it satisfying a bunch of axioms. Abstraction is beautiful, but sometimes it is useful to make things more concrete. From a mathematical point of view, this is what representation theory does for you.
The idea is simple. In fact we have already encountered it in the previous section. A representation of a group is basically a concrete realization of a group as acting on something. In this course, we will mostly deal with matrix representations; that is, we will realize the elements of an abstract group as matrices. More formally, we will define a representation as a group homomorphism from an abstract group to a subgroup of \(GL(V)\) for some finite-dimensional (complex) vector space \(V\text{.}\) After choosing a basis on \(V\text{,}\) we can represent the group elements by \(n \times n\) matrices. We will call the dimension \(n\) of the vector space the dimension of the representation.
In other words, what we are doing is shifting viewpoint. One can think of group theory as studying the group of symmetries of a given object, or theory, or what not. Instead, we now start with an abstract group, and ask the question: on what kind of objects can this group act on? Representation theory is concerned with studying all possible ways that a given group can act.
But representation theory is also more than just the study of a question of mathematical interest. In fact, in many ways it is the essence of group theory. Groups usually arise because they act on something. They are symmetries of some object, or some physical theories. Thus the elements of the group are really realized concretely, in a given context, as things that operate on some vector space, and after choosing a basis, they can be represented as matrices. So in many contexts, especially in physics, we encounter groups in terms of their representations.
In fact, many physicists think of some groups as their representations. When physicists think of the group \(SO(3)\) of rotations in three dimensions, they will often think of the corresponding \(3 \times 3\) rotation matrices. But \(SO(3)\) itself, as a group, is an abstract entity; a mathematician would think of rotations as abstract elements of \(SO(3)\text{,}\) with the abstract group operation of \(SO(3)\text{.}\) The \(3 \times 3\) matrices form in fact a three-dimensional representation of \(SO(3)\) (but there are in fact many more representations of \(SO(3)\text{!}\))
In any case, in physics groups often arise as symmetry groups of a particular system. So one has a particular physical system, and a group of symmetry acts on it to leave the physical observables invariant. However, the mathematical objects describing the physical theory (or the space of solutions) may not be invariant; for instance, while the observables in quantum field theory may be invariant, the fields themselves will not be. Thus they must transform in a certain way. This is where representation theory comes in; they will transform in some representation of the group of symmetry. This is why group theory and representation theory is so fundamental in modern physics!
Let me give a slightly more concrete example of this to end this introduction. In quantum mechanics, we may consider a Hamiltonian that is invariant under a group of transformations. A very important question then is to find how the solutions of the Schrodinger equation corresponding to this particular Hamiltonian behave under these symmetry transformations. This is answered by representation theory; we can think of the group of symmetries as linear transformations on the vector space of solutions, which is precisely the notion of a representation of the group. This can then be used to classify the eigenfunctions of the Hamiltonian according to how they transform under the symmetry group. Very powerful! So let us study representation theory!