Representation theory is a branch of mathematics that studies how abstract algebraic structures, like groups, can be represented through linear transformations of vector spaces. This allows mathematicians to translate complex algebraic problems into more manageable linear algebra problems, making it easier to analyze the properties of these structures. In particular, representation theory is often applied to the study of symmetry in various fields, including physics and chemistry. For example, it helps in understanding the symmetries of molecules and crystals, as well as in the analysis of particle physics through the representations of groups like Lie groups and finite groups.