representation theory is a very broad subject. in a nutshell, it is a systematic study of how abstract groups (or algebras) can be represented by concrete linear transformations of a vector space. a guiding example is the symmetric group on four letters, which can be thought of as the rotational symmetries of a cube. representation theory pervades diverse areas of mathematics, and even particle physics. in number theory the langlands program posits a deep connection between representations of various lie groups and representations of galois groups, through the theory of l-functions.