Representations turn abstract groups into matrices you can compute
A group representation is a homomorphism into invertible linear maps on a vector space—so each group element becomes a matrix and composition becomes multiplication. Hard group questions often reduce to linear algebra, from molecule symmetries to quantum physics.
More broadly, any homomorphism into the automorphisms of some object is a "representation"; when the object is a vector space one speaks of a linear representation, and some authors reserve "representation" for that case alone. Chemistry matches group elements to rotations and reflections of molecules. Physics tracks how a system's symmetry group acts on solution spaces of its equations. The representation space's dimension is the degree of the representation; choosing a basis identifies the target with GL(n, K). A representation is faithful when the homomorphism is injective—its kernel is only the identity.
The subject splits by the kind of group. Finite-group representations matter in crystallography and geometry; if the field's characteristic divides the group order one enters modular representation theory with sharply different behavior. For compact or locally compact groups, averages become integrals against Haar measure, feeding harmonic analysis and Pontryagin duality for commutative groups. Lie-group representations dominate applications in physics and chemistry. Linear algebraic groups over general fields need algebraic-geometry methods because the Zariski topology is coarse. Non-compact groups lack a single general theory, though semisimple and solvable cases have deep special toolkits, including Mackey theory.
The vector space itself matters: finite versus infinite dimensional, Hilbert or Banach structure, and the scalar field—complex numbers first, then reals, finite fields, p-adics. Algebraically closed fields are easier; many finite-group theorems need the characteristic not to divide the group order. Continuous representations for topological groups require the action map G × V → V to be continuous. From character tables to particle multiplets, concrete matrices keep abstract symmetry honest.
Source: Group representation
