permutation representation

Let G be a group, and S any finite set on which G acts.

That means that for any g,hG; 𝐯,𝐰S

  • g𝐯V,

  • (gh)𝐯=g(h𝐯),

  • e𝐯=𝐯.

Notice that we almost have what it takes to make S a representationPlanetmathPlanetmath of G, but S is no vector spaceMathworldPlanetmath. We can however obtain a G-module (a vector space carrying a representation of G) as follows.

Let S={𝐬1,𝐬2,,𝐬n}. And let S=[𝐬1,𝐬2,,𝐬n] be the vector space generated by S over . in other words, S is made of all formal linear combinationsMathworldPlanetmath c1𝐬1+c2𝐬2++cn𝐬n with cj. The sum is defined coordinate-wise as is scalar multiplication.

Then the action of G in S can be extended linearly to S as


and then the map ρ:GGL(S) where ρ is such that ρ(g)(𝐯)=g𝐯 makes S into a G-module. The G-module S is known as the permutation representation associated with S.

If G=Sn acts on S={𝟏,𝟐,,𝐧}, then


If σSn, the action becomes


Since S forms a basis for this space, we can compute the matrices corresponding to the defining permutationMathworldPlanetmath and we will see that the corresponding permutation matricesMathworldPlanetmath are obtained.

References. Bruce E. Sagan. The Symmetric GroupMathworldPlanetmathPlanetmath: Representations, Combinatorial Algorithms and Symmetric Functions. 2a Ed. 2000. Graduate Texts in Mathematics. Springer.

Title permutation representation
Canonical name PermutationRepresentation
Date of creation 2013-03-22 14:53:59
Last modified on 2013-03-22 14:53:59
Owner drini (3)
Last modified by drini (3)
Numerical id 6
Author drini (3)
Entry type Definition
Classification msc 20Cxx
Related topic MatrixRepresentation