group representation

Let G be a group, and let V be a vector spaceMathworldPlanetmath. A representation of G in V is a group homomorphismMathworldPlanetmath ρ:GGL(V) from G to the general linear groupMathworldPlanetmath GL(V) of invertible linear transformations of V.

Equivalently, a representation of G is a vector space V which is a G-module, that is, a (left) module over the group ringMathworldPlanetmath [G]. The equivalence is achieved by assigning to each homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ρ:GGL(V) the module structureMathworldPlanetmath whose scalar multiplication is defined by gv:=(ρ(g))(v), and extending linearly. Note that, although technically a group representationMathworldPlanetmath is a homomorphism such as ρ, most authors invariably denote a representation using the underlying vector space V, with the homomorphism being understood from context, in much the same way that vector spaces themselves are usually described as sets with the corresponding binary operationsMathworldPlanetmath being understood from context.

Special kinds of representations

(preserving all notation from above)

A representation is faithful if either of the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions is satisfied:

  • ρ:GGL(V) is injectivePlanetmathPlanetmath,

  • V is a faithful left [G]–module.

A subrepresentation of V is a subspacePlanetmathPlanetmath W of V which is a left [G]submoduleMathworldPlanetmath of V; such a subspace is sometimes called a G-invariant subspace of V. Equivalently, a subrepresentation of V is a subspace W of V with the property that

(ρ(g))(w)W for all gG and wW.

A representation V is called irreducible if it has no subrepresentations other than itself and the zero moduleMathworldPlanetmath.

Title group representation
Canonical name GroupRepresentation
Date of creation 2013-03-22 12:13:30
Last modified on 2013-03-22 12:13:30
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 20C99
Synonym representation
Related topic GeneralLinearGroup
Defines subrepresentation
Defines irreducible
Defines faithful