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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
- $\chi(e) = 1$
- $\chi(g_1g_2) = \chi(g_1)\chi(g_2)$ 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).
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