Associating Multilinear With Alternating Forms
\[V\]
be a vector space over a field \[F\]
, let \[r\]
be a positive integer.and let \[L\]
be a multilinear function (linear in each argument) \[L:V^r \rightarrow F\]
.\[L\]
is an alternating form if \[L=0\]
whenever two arguments are the same, and \[L\]
changes sign if conservative arguments are interchanged.Let
\[L\]
be an r - linear form on \[V\]
and let \[\sigma\]
be a permutation of \[1, \; 2,..., \; r\]
- there are \[n!\]
such permutations). If \[L\]
is alternating the sign of \[L\]
is defined by the number of transpositions of \[1, \; 2,..., \; r\]
needed to return the set to the natural order. Suppose that numbers \[1, \; 2,..., \; r\]
are rearranged as \[a_1, \; a_2,..., \; a_r\]
then\[L(\alpha_1 , \; \alpha_2 , \;..., \; \alpha_r)=(-1)^{e_{a_1a_2...a_r}}L(\alpha_{a1} , \; \alpha_{a_2} , \;..., \; \alpha_{a_r})\]
where
\[e_{a_1a_2...a_r}\]
is the number of transitions needed to return the set \[a_1, \; a_2,..., \; a_r\]
to the natural order. Write
\[(-1)^{e_{a_1a_2...a_r}}=sgn( \sigma )\]
then\[L(\alpha_1 , \; \alpha_2 , \;..., \; \alpha_r)=sgn( \sigma )L(\alpha_{a1} , \; \alpha_{a_2} , \;..., \; \alpha_{a_r})\]
The set of all multilinear functions forms a subspace
\[M^r(V)\]
. For each \[L \in M^r\]
define \[\prod_{\sigma } L \in M^r(V)\]
as \[\prod_{\sigma } L= \sum_{\sigma} sgn(\sigma ) L\]
.\[\prod_{\sigma } L\]
is a linear transformation from \[M^r(V)\]
onto \[A^r(V)\]
- the set of alternating r forms on \[V\]
. 
