symmetric multilinear function

Let R be a commutative ring with identityPlanetmathPlanetmathPlanetmath and M,N be unital R-modules.

Suppose that ϕ:M××MN is a multilinear map, where there are n copies of M.

Let H be a subgroupMathworldPlanetmathPlanetmath of Sn, the symmetric groupMathworldPlanetmathPlanetmath on {1,,n}, and χ:HR satisfy

  1. 1.


  2. 2.

    χ(g1g2)=χ(g1)χ(g2) for all g1,g2H

We say that ϕ is symmetricPlanetmathPlanetmath with respect to H and χ if


holds for all σH and all miM.

Now suppose that H=Sn.

If χ=1 then we say that ϕ is a symmetric multilinear function. If χ=ϵ, the sign of the permutationMathworldPlanetmath σ, we say that ϕ is a skew-symmetric multilinear function.

For example, the permanent is a symmetric multilinear function of its rows (columns).

The determinant is a skew-symmetric multilinear function of its rows (columns).

Title symmetric multilinear function
Canonical name SymmetricMultilinearFunction
Date of creation 2013-03-22 16:10:53
Last modified on 2013-03-22 16:10:53
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 11
Author Mathprof (13753)
Entry type Definition
Classification msc 13A99
Defines skew-symmetric multilinear function