Congruence of multilinear forms

Let be two n-linear forms with n 2 on finite dimensional vector spaces U and V over a field . We say that F and G are symmetrically equivalent if there exist linear bijections 1, … , n : U → V such that F(u1,…,un)=G( i1u1,…, inun) for all u1, … , un U and each reordering i1, … , in of 1, … , n. The forms are said to be congruent if 1 = = n. Let F and G be symmetrically equivalent. We prove that (i) if , then F and G are congruent; (ii) if , F = F1 Fs 0, G = G1 Gr 0, and all summands Fi and Gj are nonzero and direct-sum-indecomposable, then s = r and, after a suitable reindexing, Fi is congruent to ±Gi.