Decomposability of symmetric multilinear functions |
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Authors: | Thomas H. Pate |
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Affiliation: | Mathematics Department Auburn University Auburn, Alabama 36849, USA |
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Abstract: | Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If n denotes the symmetric group on {1,2,…,n} then we define the projection by the formula , where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If , then x1?x2? … ?xn denotes the member of Tn(V) such that for each y1 ,2,…,yn in V, and x1·x2… xn denotes . If B? Sn(V) and there exists , such that B = x1·x2…xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(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. |
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