Talk:Multilinear map
From Wikipedia, the free encyclopedia
It would be nice to define a multilinear map as a map of R-modules to an R-module that is linear in each variable as I believe this is the most general sense of the term, however I don't know of any examples of such maps where the codomain is not R itself.
Also it may be useful to define symmetric, antisymmetric and alternating multilinear functions. I believe the appropriate definitions are
A multilinear function is called symmetric if D(a_1,...,a_i,...,a_j,...,a_n) = D(a_1,...,a_j,...,a_i,...,a_n)
A multilinear function is called antisymmetric if D(a_1,...,a_i,...,a_j,...,a_n) = -D(a_1,...,a_j,...,a_i,...,a_n)
A multilinear function is called alternating if D(a_1,...,a_i,...,a_i,...,a_n) = 0
I would like someone else to verify this before I put it up. Moreover, the relationships between these seem to be that alternating implies antisymmetric (regardless of the characteristic of the field) and antisymmetric implies alternating if the characteristic is not 2. Also, symmetric = antisymmetric iff the characteristic is 2.
TooMuchMath 01:33, 29 January 2006 (UTC)
