induced representation

Let G be a group, HG a subgroupMathworldPlanetmathPlanetmath, and V a representation of H, considered as a [H]–module. The induced representationMathworldPlanetmath of ρ on G, denoted IndHG(V), is the [G]–module whose underlying vector spaceMathworldPlanetmath is the direct sumMathworldPlanetmathPlanetmathPlanetmath


of formal translatesMathworldPlanetmath of V by left cosetsMathworldPlanetmath σ in G/H, and whose multiplication operationMathworldPlanetmath is defined by choosing a set {gσ}σG/H of coset representatives and setting


where τ is the unique left coset of G/H containing ggσ (i.e., such that ggσ=gτh for some hH).

One easily verifies that the representation IndHG(V) is independent of the choice of coset representatives {gσ}.

Title induced representation
Canonical name InducedRepresentation
Date of creation 2013-03-22 12:17:33
Last modified on 2013-03-22 12:17:33
Owner djao (24)
Last modified by djao (24)
Numerical id 4
Author djao (24)
Entry type Definition
Classification msc 20C99