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symmetric multilinear function (Definition)

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. $ \chi(e) = 1$
  2. $ \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

$\displaystyle \phi(m_{\sigma(1)} , \ldots, m_{\sigma(n)}) = \chi(\sigma)\phi(m_1,\ldots,m_n)$
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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Also defines:  skew-symmetric multilinear function
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Cross-references: determinant, columns, rows, permanent, permutation, symmetric group, subgroup, map, multilinear, unital, identity, commutative ring
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This is version 8 of symmetric multilinear function, born on 2006-08-20, modified 2006-08-20.
Object id is 8269, canonical name is SymmetricMultilinearFunction.
Accessed 1204 times total.

Classification:
AMS MSC: 13A99 (Commutative rings and algebras :: General commutative ring theory :: Miscellaneous)
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