Orthogonal decompositions and canonical embeddings of multilinear alternating forms

Given k-linear forms f(i) : V-i(k) ---> F, 1 less than or equal to i less than or equal to m, define a k-linear form f = f(1) circle plus ... circle plus f(m) : (V-1 circle plus ... circle plus V-m)(k) --> F by f(u(1) ,..., u(k)) = Sigma(i)(i)(f)(P-i(u(1)),..., P-i(u(k))), where P-i : V-1 circle plus ... circle plus V-m --> V-i are projections. If a k-linear form f : V-k --> F can be expressed as above call the system of subspaces V-1 ,..., V-m an orthogonal decomposition (with respect to f). We show that for k greater than or equal to 3 such a decomposition is unique if in is maximal possible. Furthermore we prove that a nondegenerate alternating form f : V-k --> F can be always extended to h = h(1) circle plus ... circle plus h(c), where h(i) : (V-i)(k) --> F are nonzero alternating, and dim V-i = k, l less than or equal to i less than or equal to c.