Representation theory

Representation theory studies ways to realize abstract algebraic structures, especially groups and algebras, as linear transformations of vector spaces. It turns algebraic operations into matrix operations that can be studied with linear algebra.

Group: A set with an associative operation, an identity element, and an inverse for every element. Group representations assign invertible linear maps to group elements while preserving their multiplication.
Group representation: A homomorphism from a group to the invertible linear transformations of a vector space. This is the standard way representation theory converts group multiplication into composition of linear maps.
Representation of a finite group: A linear action of a finite group on a vector space. Finite groups provide a setting where character methods and decomposition are especially effective.
Permutation representation: A representation arising from a group's action by permutations of a finite set. It captures symmetry by rearranging basis vectors, a concrete special case of linear action.
Ferdinand Georg Frobenius: A German mathematician whose work helped establish the character theory of finite groups. His introduction of group characters made finite-dimensional representations easier to classify.
Vector space: A set with vector addition and scalar multiplication satisfying the axioms of linear algebra. Representations act on vector spaces, whose structure makes linear methods available.
Character theory: The study of traces of linear representations, especially characters of finite groups. Characters compress representation data into functions that often determine decompositions and equivalences.
Representation of a Lie group: A smooth or continuous action of a Lie group by linear transformations. These representations encode continuous symmetries used in geometry and mathematical physics.
Affine representation: An action by affine transformations, which combine linear maps with translations. Translations make affine actions broader than the origin-preserving linear actions central here.
Issai Schur: A German mathematician known for foundational contributions to representation theory and invariant theory. Schur's lemmas became fundamental tools for understanding maps between irreducible representations.

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