Congruence of multilinear forms

Abstract

Let F : U × ⋯ × U → K , G : V × ⋯ × V → K be two n-linear forms with n ⩾ 2 on finite dimensional vector spaces U and V over a field K . We say that F and G are symmetrically equivalent if there exist linear bijections ϕ 1, … , ϕ n : U → V such that F ( u 1 , … , u n ) = G ( ϕ i 1 u 1 , … , ϕ i n u n ) for all u 1, … , u n ∈ U and each reordering i 1, … , i n of 1, … , n. The forms are said to be congruent if ϕ 1 = ⋯ = ϕ n . Let F and G be symmetrically equivalent. We prove that (i) if K = C , then F and G are congruent; (ii) if K = R , F = F 1 ⊕ ⋯ ⊕ F s ⊕ 0, G = G 1 ⊕ ⋯ ⊕ G r ⊕ 0, and all summands F i and G j are nonzero and direct-sum-indecomposable, then s = r and, after a suitable reindexing, F i is congruent to ± G i .