Preface Motivation
ΒΆSymmetry is the language of modern physics. When confronted with a physical system, the theoretical physicist's main tool to formulate a mathematical description of the system is to study its symmetries. Those correspond to transformations of the physical system that leaves the physics invariant.
The mathematical language for studying transformations and symmetries is group theory. This is why group theory is omnipresent in modern theoretical physics. We study what kind of transformations leave a physical system invariant; those transformations generally form an abstract group (the symmetry group of the system), and then we use the language of group theory to describe the physical system.
Groups are defined abstractly. But what they consist in is a set of symmetries (of an object, a theory, etc.) that is closed under composition and contains inverse operations. Most symmetries of nature are of that sort, which is why group theory appears universally.
But in fact in physics what we are interested in is how groups act on something: an object, a theory. This is what representation theory is about: it represents the elements of a group as acting on something. While in geometry, one may ask what the symmetries of a particular object are, in representation theory, the question is turned around: given a group, what kind of objects does it act on? This is the central theme of representation theory.
In particular, in physics groups of symmetries act on the space of solutions of a physical system. This is given by a representation of the symmetry group. Understanding the details of this representation tells us a lot about the properties of the possible solutions (or particles, fields, etc.)
Therefore, by studying group theory and representation theory, we will learn a lot about physical entities. We will see how the spin of a particle arises naturally in terms of representations, how particles in the Standard Model transform according to representations of the gauge group \(SU(3) \times SU(2) \times U(1)\text{,}\) how these representations naturally give rise to the idea of Grand Unified Theories, etc. Cool stuff!