5.8 Tensor Decomposable Tensor Structure

Tensor Decomposable Tensor Structure is the body of theory describing decomposable tensors — elements of a tensor product, or of a related multilinear construction such as an exterior or symmetric power, that can be factored (decomposed) into a single product of vectors from the constituent spaces — together with the algebraic variety these elements trace out, the polynomial relations characterizing them, and the geometric objects they parametrize in more structured settings.

The Decomposable Property

For vector spaces V1, V2, …, Vn over a field F and their tensor product T, an element t of T is decomposable if it factors as

The term "decomposable" emphasizes the reverse viewpoint from "simple": rather than describing t as a single elementary building block, it describes t as something that comes apart cleanly into n independent pieces, one per factor space, with no residual entanglement between them.

Decomposability in the Exterior Algebra

The term decomposable tensor is used with particular precision in the exterior algebra, where it carries direct geometric meaning connecting multilinear algebra to the study of linear subspaces.

Decomposable p-Vectors

An element ω of the p-th exterior power ⋀ᵖV of a vector space V is decomposable if it can be written as a wedge product of p vectors,

Every decomposable p-vector, up to nonzero scalar multiple, corresponds to exactly one p-dimensional linear subspace of V, namely the span of w1, …, wp; this correspondence is the algebraic foundation of the Grassmannian, the space parametrizing all p-dimensional subspaces of V.

Plücker Relations

A general element of ⋀ᵖV, expressed in coordinates relative to a basis, is decomposable exactly when its coordinates satisfy the Plücker relations, a specific system of quadratic polynomial equations. These relations cut out the Grassmannian, embedded via the Plücker embedding, as an algebraic variety inside the projectivization of ⋀ᵖV, giving the decomposability condition a concrete, checkable algebraic form.

Decomposability in the Symmetric Algebra

A parallel structure exists in the symmetric setting, connecting decomposable symmetric tensors to points rather than subspaces.

Decomposable Symmetric Tensors and the Veronese Variety

An element of the p-th symmetric power of V is decomposable if it equals vᵖ (the p-fold symmetric product of a single vector v with itself). Under the standard identification of symmetric tensors with homogeneous polynomials, decomposability corresponds to the polynomial being a perfect p-th power of a linear form. The decomposable symmetric tensors, up to scale, trace out the Veronese variety inside the projectivization of the symmetric power.

The Decomposable Locus as an Algebraic Variety

Across all of these settings — plain tensor products, exterior powers, and symmetric powers — the decomposable elements form a common geometric object: a variety cut out by homogeneous quadratic equations, closed under scaling but not under addition.

Cone Structure

The decomposable locus is a cone: if t is decomposable, so is λt for any scalar λ, since scaling one factor scales the whole product. It is not a linear subspace, since the sum of two decomposable elements is generically not decomposable — the same obstruction that makes tensor rank a nontrivial invariant.

Codimension and Genericity

In each of these settings, the decomposable locus has strictly smaller dimension than the ambient space once the relevant exponent (the number of tensor factors, or the degree p) exceeds one, meaning a generic (randomly chosen) element of the ambient space is not decomposable. This codimension gap is what makes decomposability a genuine constraint rather than a property held by almost every element.

Structural Consequences

Recognizing the decomposable locus as an algebraic variety, rather than treating decomposability as a purely combinatorial property, brings the tools of algebraic geometry to bear on tensor decomposition problems.

Secant Varieties and Rank

The variety of tensors of rank at most r is, in this geometric language, the r-th secant variety of the decomposable locus — the union of all linear spans of r points on the decomposable variety. Questions about generic tensor rank, maximal rank, and the failure of naive dimension counts to predict rank (so-called defective secant varieties) are studied directly through the geometry of the underlying decomposable locus.

Tangent Spaces and Local Approximation

The tangent space to the decomposable locus at a given decomposable element describes the directions in which that element can be infinitesimally perturbed while remaining, to first order, close to decomposable. This local, differential-geometric picture underlies iterative numerical algorithms for best low-rank tensor approximation, which repeatedly move along or near the decomposable locus.

Illustrative Diagram

The curved path traces the decomposable elements as a lower-dimensional variety inside the flat ambient space, illustrating why decomposability is a special, non-generic condition rather than a property shared by every element.