Representation theory is the study of how abstract algebraic structures (like groups, algebras, Lie groups, Lie algebras) can be represented concretely as linear transformations (matrices) acting on vector spaces.

Concisely, homomorphism into linear transformations!

Let’s talk about Lorentz representation, Spinor representation and Lie Algebra version to grasp this concept.

there are three rotations Ji and three boosts Ki

There is global symmetry and local symmetry, locally, due to curvature, field also transform. Gauge theory is just matrix-valued calculus enforcing local symmetry consistency.