permutation operator

Let V be a vector spaceMathworldPlanetmath over a field. Let σSn, the symmetric group on {1,,n} and define a multilinear map ϕ:V××VVn=VVn times by


Then by the universalPlanetmathPlanetmathPlanetmath factorization property ( for a tensor productPlanetmathPlanetmathPlanetmath ( there is a unique linear map P(σ):VnVn such that P(σ)=ϕ. Then of course,


P(σ) is called the permutation operator associated with σ.

1 Properties

  1. 1.


  2. 2.

    P(e)=I , where I is the identity mapping on Vn

  3. 3.

    P(σ) is nonsingular and P(σ)-1=P(σ-1)

Title permutation operator
Canonical name PermutationOperator
Date of creation 2013-03-22 16:15:38
Last modified on 2013-03-22 16:15:38
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type Definition
Classification msc 15A04