example of induced representation

To understand the definition of induced representationMathworldPlanetmath, let us work through a simple example in detail.

Let G be the group of permutationsMathworldPlanetmath of three objects and let H be the subgroupMathworldPlanetmathPlanetmath of even permutationsMathworldPlanetmath. We have


Let V be the one dimensional representation of H. Being one-dimensional, V is spanned by a single basis vector v. The action of H on V is given as


Since H has half as many elements as G, there are exactly two cosets, σ1 and σ2 in G/H where


Since there are two cosets, the vector spaceMathworldPlanetmath of the induced representation consists of the direct sumMathworldPlanetmathPlanetmathPlanetmath of two formal translatesMathworldPlanetmath of V. A basis for this space is {σ1v,σ2v}.

We will now compute the action of G on this vector space. To do this, we need a choice of coset representatives. Let us choose g1=e as a representative of σ1 and g2=(ab) as a representative of σ2. As a preliminary step, we shall express the productPlanetmathPlanetmath of every element of G with a coset representative as the product of a coset representative and an element of H.


We will now compute of the action of G using the formulaMathworldPlanetmathPlanetmath g(σv)=τ(hv) given in the definition.


Here the square brackets indicate the coset to which the group element inside the brackets belongs. For instance, [(ac)g2]=[(ac)(ab)]=[(acb)]=σ1 since (acb)σ1.

The results of the calculation may be easier understood when expressed in matrix form

e    (1001)
(ab)    (0110)
(bc)    (0exp(2πi/3)exp(4πi/3)0)
(ac)    (0exp(4πi/3)exp(2πi/3)0)
(abc)    (exp(2πi/3)00exp(4πi/3))
(acb)    (exp(4πi/3)00exp(2πi/3))

Having expressed the answer thus, it is not hard to verify that this is indeed a representation of G. For instance, (acb)(ac)=(bc) and

Title example of induced representation
Canonical name ExampleOfInducedRepresentation
Date of creation 2013-03-22 14:35:43
Last modified on 2013-03-22 14:35:43
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Example
Classification msc 20C99