Let V be a vector spaceMathworldPlanetmath over a field F. Let n be an integer, where n<char(F) if char(F)0. Let Sn be the symmetric groupMathworldPlanetmathPlanetmath on {1,,n}. The linear operator S:VnVn defined by:


is called the symmetrizer. Here P(σ) is the permutation operator. It is clear that P(σ)S=SP(σ)=S for all σSn.

Let S be the symmetrizer for Vn. Then an order-n tensor A is symmetricPlanetmathPlanetmath ( if and only S(A)=A.

If A is then


If S(A)=A then


for all σSn, so A is .

The theorem says that a is an eigenvectorMathworldPlanetmathPlanetmathPlanetmath of the linear operator S corresponding to the eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath 1. It is easy to verify that S2=S, so that S is a projectionPlanetmathPlanetmath onto Sn(V).

Title symmetrizer
Canonical name Symmetrizer
Date of creation 2013-03-22 16:15:44
Last modified on 2013-03-22 16:15:44
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 8
Author Mathprof (13753)
Entry type Definition
Classification msc 15A04