symmetric multilinear function
Let $R$ be a commutative ring with identity^{} and $M,N$ be unital $R$modules.
Suppose that $\varphi :M\times \mathrm{\cdots}\times M\to N$ is a multilinear map, where there are $n$ copies of $M$.
Let $H$ be a subgroup^{} of ${S}_{n}$, the symmetric group^{} on $\{1,\mathrm{\dots},n\}$, and $\chi :H\to R$ satisfy

1.
$\chi (e)=1$

2.
$\chi ({g}_{1}{g}_{2})=\chi ({g}_{1})\chi ({g}_{2})$ for all ${g}_{1},{g}_{2}\in H$
We say that $\varphi $ is symmetric^{} with respect to $H$ and $\chi $ if
$$\varphi ({m}_{\sigma (1)},\mathrm{\dots},{m}_{\sigma (n)})=\chi (\sigma )\varphi ({m}_{1},\mathrm{\dots},{m}_{n})$$ 
holds for all $\sigma \in H$ and all ${m}_{i}\in M$.
Now suppose that $H={S}_{n}$.
If $\chi =1$ then we say that $\varphi $ is a symmetric multilinear function. If $\chi =\u03f5$, the sign of the permutation^{} $\sigma $, we say that $\varphi $ is a skewsymmetric multilinear function.
For example, the permanent is a symmetric multilinear function of its rows (columns).
The determinant is a skewsymmetric multilinear function of its rows (columns).
Title  symmetric multilinear function 

Canonical name  SymmetricMultilinearFunction 
Date of creation  20130322 16:10:53 
Last modified on  20130322 16:10:53 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  11 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 13A99 
Defines  skewsymmetric multilinear function 