representation theory

Representation theory is a branch of mathematics that studies how abstract algebraic structures, like groups, can be expressed through matrices and linear transformations. Essentially, it connects algebra with geometry by showing how these structures can act on vector spaces. This framework helps in understanding symmetrical patterns and properties in various fields, including physics and chemistry. By interpreting algebraic entities visually and numerically, representation theory provides powerful tools for analyzing the behavior of complex systems and solving problems across multiple disciplines.