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 characteristic zero (or when the field’s characteristic does not divide the group order), Maschke’s theorem guarantees complete reducibility. Irreducible representations have endomorphism rings that are simple, by Schur’s lemma. The traces of representation matrices, called characters, form a powerful toolkit: character theory uses orthogonality relations to distinguish and classify representations, often summarized in character tables.

Beyond groups, representation theory extends to algebras, Lie algebras, and Lie groups. Representations of Lie algebras are studied via modules over their universal enveloping algebras, with highest-weight theory classifying finite-dimensional representations of semisimple Lie algebras. Geometric and categorical perspectives have expanded the field, linking representation theory to algebraic geometry, number theory, and mathematical physics through objects such as derived categories and geometric representation theory.

Applications span physics, where symmetry governs quantum states; chemistry, where group representations explain molecular orbitals and spectra; and number theory, where representations of Galois groups and automorphic forms illuminate arithmetic questions. Representation theory thus provides a unifying framework across disciplines by translating symmetry and structure into linear action.