Group representation

A group representation is a homomorphism from a group to the group of invertible linear transformations of a vector space. It expresses abstract group elements as linear operators while preserving the group operation.

Group: A set with an associative operation, an identity element, and an inverse for every element. A representation starts with a group and translates its operation into composition of linear maps.
Representation matrix: A matrix describing how a group element acts in a chosen basis of a representation space. Choosing a basis turns each represented transformation into a concrete matrix.
Representation theory of finite groups: The study of linear representations of groups with finitely many elements. Finite groups provide a setting where representations often decompose into irreducibles.
Group action: A rule assigning each group element a transformation of a set while respecting the group operation. A group action needs only a set, whereas a representation requires linear transformations.
Vector space: A set supporting vector addition and scalar multiplication over a field, subject to axioms. The representation acts on this space through invertible linear transformations.
Invariant subspace: A subspace mapped into itself by every operator in a specified family. Such subspaces show when a representation contains smaller representations.
Lie group representation: A representation of a Lie group that is compatible with its smooth structure. Continuous symmetries are modeled by representations of Lie groups.
Permutation representation: A representation arising from a group action by permutations of a basis. It is a special representation whose matrices permute basis vectors.
Group homomorphism: A function between groups that preserves their operation. The defining map must preserve multiplication as composition of transformations.
Irreducible representation: A nonzero representation with no proper nonzero invariant subspaces. Irreducibility identifies representations that cannot be decomposed further.

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