15.20.2 Tensor Symmetric Tensor Form Role

Tensor Symmetric Tensor Form Role is the identification of a symmetric tensor with a symmetric multilinear form, a function taking several vector arguments rather than a single one, and it is the perspective, historically prior to and distinct from the single-argument polynomial perspective, that names symmetric tensors of every order collectively as "forms" throughout classical invariant theory.

The Multilinear Functional Perspective

Forms as Multi-Argument Functions

A symmetric tensor T of order d on a vector space V can be regarded as a function taking d vector arguments, linear in each argument separately, and unchanged when its arguments are permuted in any order:

for every permutation sigma. This is the Form Role of the tensor: rather than being converted, as in the Tensor Symmetric Tensor Polynomial Role, into a single-argument homogeneous polynomial by restricting to d copies of one vector, T is retained here in its full, genuinely multi-argument shape, matching precisely the definition of a symmetric bilinear form generalized from order two, as introduced under the Bilinear Form Relation, to arbitrary order d.

Classical Terminology: Binary, Ternary, and Higher Forms

Classical invariant theory names symmetric tensors, in this multilinear functional guise together with their associated single-argument polynomial restriction, according to the dimension of the underlying vector space: a "binary form" is a symmetric tensor on a two-dimensional space, a "ternary form" on a three-dimensional space, and so on, with the order d of the tensor supplying the further qualifier of "quadratic," "cubic," "quartic," and higher, matching the degree of the associated polynomial. The Form Role is precisely what this classical terminology refers to when it speaks of "forms" as the primary object of study, prior to and independent of the coordinate-based polynomial expressions used to compute with them.

Distinguishing the Form Role from the Polynomial Role

Multi-Argument versus Single-Argument Evaluation

Although the Form Role and the Polynomial Role describe the same underlying tensor T, they emphasize different modes of evaluation: the Form Role evaluates T on d potentially distinct vectors, retaining full multilinear information, while the Polynomial Role evaluates T on d copies of a single vector, compressing this information into a homogeneous polynomial. The Tensor Quadratic Form Polarization Relation is precisely the bridge recovering the full multi-argument form from its single-argument polynomial restriction, and it is stated in exactly the terms made available by holding both roles in view simultaneously.

Complementary Uses in Practice

The Polynomial Role is generally more convenient for questions concerning rank, decomposition, and the geometry of the Veronese variety, since these are naturally phrased in terms of the single associated polynomial and its vanishing or factorization behavior; the Form Role, by contrast, is more natural whenever the tensor is to be paired against several distinct, meaningfully different vectors at once, as occurs when a symmetric tensor is contracted against a mix of decomposition vectors and arbitrary test vectors during Reconstruction, or when studying the bilinear pairing between a tensor and lower-order tensors via partial contraction.

Structural Consequences of the Form Role

Symmetric Forms as a Graded Family

Collecting symmetric forms of every order d on a fixed vector space V produces exactly the symmetric algebra discussed under the Polynomial Role, confirming that the Form Role and the Polynomial Role, while notationally and conceptually distinct, describe the same underlying graded algebraic structure; the Form Role simply insists on remembering, at each stage, the full multi-argument functional rather than immediately restricting to the diagonal.

Classical Invariant Theory of Forms

The historical development of invariant theory, concerned with finding polynomial expressions in the coefficients of a form that remain unchanged under a linear change of variables, is conducted almost entirely in the Form Role's language: invariants and covariants of binary and ternary forms are defined directly in terms of the coefficients of the associated symmetric tensor, and classical results, such as the classification of invariants of binary cubic and quartic forms, are theorems about symmetric tensors understood specifically through this multilinear Form Role, later reinterpreted through the modern lens of secant varieties, apolarity, and the Alexander-Hirschowitz classification used throughout contemporary treatments of Tensor Symmetric Decomposition Structure.

Position Within the Broader Algebraic Role

A Named Perspective, Not a Separate Object

The Form Role is not a distinct mathematical structure from the symmetric tensor itself, but rather a chosen way of presenting and evaluating it, standing alongside the Polynomial Role as one of the two principal perspectives comprising the broader Symmetric Tensor Algebraic Role; recognizing when the Form Role's multi-argument evaluation is more natural than the Polynomial Role's single-argument restriction is largely a matter of matching the perspective to the specific structural or historical question being asked about a given symmetric tensor.