15.6 Tensor Symmetric Multilinear Form Structure

Tensor Symmetric Multilinear Form Structure is the generalization of the symmetric bilinear form structure to a totally symmetric tensor of arbitrary rank n, under which the tensor defines a map taking n vector arguments and returning a scalar, linear in each argument separately, and invariant under every permutation of the n arguments. Where the bilinear case involves only two argument slots and a single possible exchange, the multilinear case involves n argument slots and the full symmetric group of n! permutations, every one of which must leave the value of the form unchanged for the tensor to qualify as totally symmetric.

This structure extends each piece of the bilinear picture: the argument pair becomes an argument tuple whose entries can be freely permuted, the slot exchange operator becomes an action of the full permutation group rather than a single transposition, the quadratic relation becomes a relation between the multilinear form and the associated degree-n homogeneous polynomial obtained by setting all arguments equal, and the matrix representation becomes a higher-dimensional array that is invariant under permutation of its indices rather than a two-dimensional symmetric matrix.

The Multilinear Map and Its Symmetry

Definition on n Arguments

A totally symmetric tensor T of rank n with components T_{i1...in} defines a multilinear form:

which reduces to the bilinear form when n equals two.

Full Permutation Invariance

Total symmetry of T means B(v_1, ..., v_n) equals B(v_σ(1), ..., v_σ(n)) for every permutation σ of the labels 1 through n, exactly as the equality constraint on the components T_{i1...in} demands invariance under permuting the index positions.

Generalized Slot Exchange

Action of the Symmetric Group

Where the bilinear case has a single nontrivial slot exchange, the multilinear case admits an action of the full symmetric group on n letters, with each permutation σ inducing an operator E_σ on multilinear forms defined by E_σ(B)(v_1, ..., v_n) = B(v_σ(1), ..., v_σ(n)). Total symmetry of B is the statement that E_σ(B) equals B for every σ in the group, not merely for a single transposition.

Generation by Transpositions

Because every permutation can be written as a product of adjacent transpositions, invariance under every single transposition of neighboring arguments is sufficient to guarantee invariance under the entire symmetric group, so checking total symmetry reduces to checking pairwise slot exchange invariance across every pair of adjacent argument positions.

Generalized Quadratic Relation

The Associated Homogeneous Polynomial

Setting all n arguments of a totally symmetric multilinear form equal to a single vector v produces a homogeneous degree-n polynomial:

generalizing the quadratic form obtained in the rank-2 case, with degree n rather than degree two.

Generalized Polarization

Just as the bilinear form is recoverable from its quadratic form by polarization, a totally symmetric multilinear form of rank n is recoverable from its degree-n homogeneous polynomial by a generalized polarization identity involving a signed sum over evaluations of Q at various integer combinations of the n vector arguments, valid whenever the underlying field allows division by n factorial.

Generalized Matrix Representation

Symmetric Array in a Fixed Basis

Fixing a basis converts the rank-n totally symmetric tensor into an n-dimensional array of numbers, invariant under permutation of its n index positions, generalizing the two-dimensional symmetric matrix of the bilinear case. This array can no longer be visualized as rows and columns for n greater than two, but the invariance under index permutation plays exactly the same structural role as the row-column symmetry of a symmetric matrix.

Independent Components in the General Case

The number of independent entries in this array follows the same orbit-counting formula used for the general symmetric component equality constraint, namely the count of multisets of size n drawn from d values, reducing to the familiar count of upper-triangular-plus-diagonal entries of a symmetric matrix when n equals two.

Relation to the Bilinear Case

Bilinear Structure as the Rank-2 Instance

Every structural element of the symmetric multilinear form, the permutation-invariant map, the group action generalizing slot exchange, the homogeneous polynomial generalizing the quadratic form, and the permutation-invariant array generalizing the symmetric matrix, specializes exactly to the corresponding bilinear structure when n is set to two, confirming that the bilinear form structure is not a separate theory but the smallest nontrivial case of the general symmetric multilinear form structure.

Increasing Complexity With Rank

While the bilinear case is fully captured by classical matrix theory, including diagonalization and signature, the multilinear case for rank greater than two generally lacks such a complete classification, and questions about the structure of a totally symmetric tensor of rank three or higher, such as its decomposition into simpler symmetric pieces, require tools beyond ordinary linear algebra.