representationteorian
Representation theory is a branch of mathematics that studies abstract algebraic structures by realizing their elements as linear transformations of vector spaces. A representation of a group G on a vector space V over a field F is a homomorphism from G to GL(V). More generally, algebras and rings can be represented by homomorphisms into the endomorphism algebra End(V). Representations translate algebraic questions into problems in linear algebra and module theory, providing concrete models to analyze structure and symmetry.
For finite groups, a central aim is to decompose representations into irreducible components. Over fields of
Beyond groups, representation theory extends to algebras, Lie algebras, and Lie groups. Representations of Lie algebras
Applications span physics, where symmetry governs quantum states; chemistry, where group representations explain molecular orbitals and