15.21.3 Tensor Symmetric Product Boundary

Tensor Symmetric Product Boundary is the account of where the well-behaved elementary properties of the symmetric product, commutativity, multilinearity, and its clean behavior on single vectors, cease to extend naively once the operation is applied to general symmetric tensors rather than to individual vectors, particularly with respect to how rank behaves under the product.

The Product Is Well-Behaved on Generators

Pure Powers Multiply to Pure Powers

As established under the Symmetric Product Notation, the symmetric product of k copies of a single vector v recovers the pure power form v raised to the tensor power k, and more generally, the symmetric product of a pure power form u raised to the power p with a pure power form v raised to the power q is again a simple, easily described object whenever u and v are proportional, since it reduces to a pure power of the common vector; on generators, the product behaves exactly as naive multiplicative intuition would suggest.

Where the Naive Intuition Stops Applying

Once the two factors are general elements of S^p V and S^q V, rather than pure power forms specifically, the resulting element of S^{p+q} V need not itself be a pure power form, nor does its symmetric rank relate to the ranks of the two factors by any simple additive or multiplicative formula; the Symmetric Product Boundary is precisely the recognition that the elementary, easily verified behavior on generators does not extend to a comparably simple statement for arbitrary elements of the symmetric algebra.

Rank Behavior Under the Product

Sub-Multiplicativity and Its Failure to Be an Equality

If T has symmetric rank r, decomposing as a sum of r pure power forms, and S has symmetric rank s, then the symmetric product of T and S, expanded by multilinearity into a sum of r times s symmetric products of individual pure power form pairs, gives an upper bound of r times s on the symmetric rank of the product; this upper bound, however, is frequently not tight, and determining exactly when it fails to be tight, and by how much, is a genuinely difficult question with no complete general answer, marking a clear limit on how far the elementary product formula extends.

Connection to Additivity Questions for Tensor Rank

The analogous question for ordinary (non-symmetric) tensor rank, whether the rank of a suitably combined pair of tensors equals the sum of their individual ranks, was long conjectured to hold in a related form (Strassen's additivity conjecture) before being shown false by an explicit counterexample; this resolution serves as a cautionary parallel for the Symmetric Product Boundary, illustrating that intuitive multiplicative or additive rank formulas, plausible from the behavior of the product on simple generators, cannot be assumed to hold for general elements of the symmetric algebra without independent proof, and in the symmetric setting the analogous additivity questions remain, in many regimes, only partially resolved.

Boundary at the Level of Underlying Vector Spaces

The Product Requires a Single Common Space

The Symmetric Product Notation is defined for vectors, and more generally elements, drawn from a single, fixed vector space V; there is no direct symmetric product combining an element of S^p V with an element of S^q W for a genuinely different space W without first embedding both into a common ambient space, such as the direct sum of V and W, at which point the product is really being taken within S^{p+q} of that direct sum rather than within any construction native to V and W separately. This marks a boundary at which the Symmetric Product Notation must be recognized as inherently single-space, with any apparent extension across multiple spaces silently invoking this direct-sum embedding.

Interaction with the Direct Sum Decomposition

When V is itself a direct sum of two subspaces, the symmetric power of V decomposes accordingly into a direct sum of products of symmetric powers of the two summands, weighted by binomial-type multiplicities; while this decomposition is completely well understood and poses no difficulty in itself, correctly tracking how a given symmetric product distributes across the resulting direct sum pieces, and how rank interacts with this distribution, is a further source of the same kind of subtlety already identified above, since the rank of a tensor supported across multiple summands is generally not simply related to the ranks of its components in each summand.

Polynomial Ring Perspective on the Boundary

Unique Factorization Concerns a Different Notion Entirely

Under the Polynomial Role, the symmetric product corresponds to ordinary polynomial multiplication, and the polynomial ring is a unique factorization domain, so every homogeneous polynomial factors uniquely, up to order and scalar, into irreducible factors; this factorization, however, answers a different question from the rank-additivity question raised above, since the irreducible factorization of a polynomial and its expression as a sum of the fewest possible powers of linear forms (its Waring rank) are unrelated notions of "simplicity," and conflating unique factorization with any statement about how symmetric rank behaves under the symmetric product is a further, easily made error the Symmetric Product Boundary is meant to guard against.

Practical Guidance at the Boundary

Verify Rank Claims Directly Rather Than by Analogy

Because no general formula governs how symmetric rank behaves under the symmetric product outside of the pure-power-form generators, any claim about the rank of a product of two specific symmetric tensors, arising for instance while composing decompositions built independently through Reconstruction on two separate factors, must be verified directly, by exhibiting an explicit decomposition of the product and confirming its minimality through apolarity or catalecticant rank bounds, rather than inferred by analogy from the ranks of the two original factors.