15.6.3 Tensor Symmetric Multilinear Component Constraint

The Tensor Symmetric Multilinear Component Constraint enforces symmetry and multilinearity in tensor components, structuring algebraic relationships.

Tensor Symmetric Multilinear Component Constraint is the index-level condition that must hold on the components of a rank-n tensor in order for the multilinear form it defines to satisfy the permutation rule of total symmetry. While the permutation rule is stated as a property of the map from argument tuples to scalars, the component constraint is the equivalent statement written directly in terms of the tensor's numerical entries, requiring that any two components whose index tuples are related by a permutation carry an identical value.

This constraint is the bridge between the abstract multilinear form structure and the concrete array of numbers used to represent a symmetric tensor in a chosen basis: it is what must be verified, or enforced, at the level of components in order to guarantee that the resulting multilinear form behaves as a symmetric map on its argument set. Because the constraint is stated purely in terms of index tuples and their permutations, it can be checked or imposed without reference to any vector space or basis choice beyond the labeling of indices.

Statement of the Constraint

Componentwise Equality Across an Orbit

For a rank-n tensor T with components T_{i1...in}, each index ranging over 1 through d, the multilinear component constraint requires:

T_{i_{1} \dots i_{n}} = T_{i_{σ (1)} \dots i_{σ (n)}}

for every permutation σ of the positions 1 through n, applied simultaneously to every index tuple (i_1, ..., i_n) in the tensor's full range.

Constraint as a System of Linear Equations

Viewed algebraically, this constraint is a homogeneous system of linear equations relating the d^n unknown component values, one equation for each pair of index tuples related by a nontrivial permutation; the solution space of this linear system is precisely the subspace of totally symmetric tensors within the full tensor space.

Deriving the Constraint From the Permutation Rule

Basis Vector Substitution

The multilinear permutation rule for the associated form B, applied specifically to arguments chosen from the basis vectors e_1 through e_d, reads B(e_{i1}, ..., e_{in}) equals B(e_{iσ(1)}, ..., e_{iσ(n)}) for every permutation σ. Since B evaluated on basis vectors returns exactly the tensor's components, this substitution converts the abstract permutation rule directly into the componentwise equality constraint.

Equivalence in Both Directions

Conversely, given a tensor whose components already satisfy the constraint, multilinearity of the associated form extends the componentwise equality to arbitrary vector arguments by expanding each argument in the chosen basis and applying linearity in each slot, so the component constraint and the permutation rule on the multilinear form are logically equivalent, each implying the other.

Structure of the Constraint's Solution Set

Orbits as the Natural Unit of the Constraint

The constraint partitions the full set of index tuples into permutation orbits, and within each orbit, every one of the constraint's equations reduces to a chain of equalities linking all members of that orbit to a single common value; solving the constraint amounts to selecting one free parameter per orbit rather than one free parameter per individual index tuple.

Dimension of the Symmetric Subspace

Because each orbit contributes exactly one degree of freedom once the constraint is imposed, the dimension of the subspace of tensors satisfying the constraint equals the number of orbits, matching the multiset-counting formula used throughout the study of symmetric tensors and confirming that the component constraint, the independent selection, and the redundancy reduction all describe the same underlying reduction in degrees of freedom from three complementary angles.

Enforcing the Constraint on an Arbitrary Tensor

Symmetrization as Constraint Satisfaction

Given an arbitrary tensor S that does not satisfy the constraint, the symmetrization operator produces a tensor T that does, by averaging S's components over every permutation of the index positions:

T_{i_{1} \dots i_{n}} = \frac{1}{n!} \sum_{σ}^{} S_{i_{σ (1)} \dots i_{σ (n)}}

and re-permuting the resulting T merely reorders the terms of this sum, leaving its value unchanged and confirming that T satisfies the component constraint exactly.

Constraint Already Satisfied Is Left Unchanged

When the input tensor already satisfies the constraint, every term in the symmetrization sum is equal, so the average equals that common value, and the symmetrization operator acts as the identity on tensors already lying in the constrained subspace, confirming that symmetrization is a projector onto the set of tensors obeying the multilinear component constraint.