# symmetric multilinear function

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

Suppose that $\phi:M\times\cdots\times M\to N$ is a multilinear map, where there are $n$ copies of $M$.

Let $H$ be a subgroup of $S_{n}$, the symmetric group on $\{1,\ldots,n\}$, and $\chi:H\to R$ satisfy

1. 1.

$\chi(e)=1$

2. 2.

$\chi(g_{1}g_{2})=\chi(g_{1})\chi(g_{2})$ for all $g_{1},g_{2}\in H$

We say that $\phi$ is symmetric with respect to $H$ and $\chi$ if

 $\phi(m_{\sigma(1)},\ldots,m_{\sigma(n)})=\chi(\sigma)\phi(m_{1},\ldots,m_{n})$

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 $\phi$ is a symmetric multilinear function. If $\chi=\epsilon$, the sign of the permutation $\sigma$, we say that $\phi$ 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 SymmetricMultilinearFunction 2013-03-22 16:10:53 2013-03-22 16:10:53 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 13A99 skew-symmetric multilinear function