Decomposability of symmetric multilinear functions

Abstract

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T n ( V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S n ( V) denote the subspace of T n ( V) whose members are the fully symmetric members of T n ( V). If L n denotes the symmetric group on {1,2,…, n} then we define the projection P L : T n(V) → S n(V) by the formula (n!) −1Σ σ ϵ L n P σ , where P σ : T n ( V) → T n ( V) is defined so that P σ ( A)( y 1, y 2,…, y n = A( y σ(1) , y σ(2) ,…, y σ( n) ) for each A ϵ T n ( V) and y i ϵ V, 1 ⩽ i ⩽ n. If x i ϵ V ∗, 1 ⩽ i ⩽ n , then x 1⊗ x 2⊗ … ⊗ x n denotes the member of T n ( V) such that (x 1⊗x 2· ⊗ ⋯ ⊗x n)(y 1,y 2,…,y n) = П n i=1x i(y i) for each y 1 , 2,…, y n in V, and x 1· x 2… x n denotes P L (x 1⊗x 2⊗ … ⊗x n) . If B ϵ S n ( V) and there exists x i ϵ V ∗, 1 ⩽ i ⩽ n , such that B = x 1· x 2… x n , then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S n ( V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.