A group representation is a mathematical group action on a vector space.

A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2 = {0, 1} has a representation ϕ by ϕ(0) v = v and ϕ(1) v = - v. A representation is a group homomorphism ϕ:G->G L(V). Most groups have many different representations, possibly on different vector spaces. For example, the symmetric group S_3 = {e, (12), (13), (23), (123), (132)} has a representation on R by ϕ_1(σ) v = sgn(σ) v, where sgn(σ) is the permutation symbol of the permutation σ.

group | irreducible representation | Lie algebra representation | multiplicative character | orthogonal group representations | Peter-Weyl theorem | primary representation | representation tensor product | Schur's lemma | semisimple Lie group | vector space

college level