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 thereare $n$ 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 $\chi =ϵ$, the sign of the permutation $\sigma$, we say that$\varphi$ 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).