Decomposable symmetric tensors

Larry Jean Cummings

Vol. 35 (1970), No. 1, 65–77

DOI: 10.2140/pjm.1970.35.65

Abstract

A k-field is a field over which every polynomial of degree less than or equal to k splits completely. The main theorem characterizes the maximal decomposable subspaces of the k-th symmetric space ∨_kV , where V is finite-dimensional vector space over an infinite k-field. They come in three forms: (1) {x₁ ∨⋯ ∨ x_k : x_k ∈ V },x₁,⋯,x_k−1 fixed; (2) ⟨a,b⟩_k = {x₁ ∨⋯ ∨ x_k : x_i ∈⟨a,b⟩}; and (3) {x₁ ∨⋯ ∨ x_k−r ∨⟨a,b⟩_(r′)},x₁,⋯ , Xk-r fixed; where a and b are linearly independent vectors in V and ⟨a,b⟩ is the subspace spanned by a and b.

Mathematical Subject Classification

Primary: 15.85

Milestones

Received: 17 March 1969

Published: 1 October 1970