15.4.5 Tensor Symmetric Component Reconstruction Rule

Tensor Symmetric Component Reconstruction Rule is the explicit procedure by which the full array of components of a symmetric tensor is recovered from a stored independent selection together with the symmetric component equality constraint. Given only the values assigned to a set of canonical representative index tuples, the reconstruction rule specifies, for any arbitrary index tuple, which representative to consult and how to read off the correct value, guaranteeing that the reconstructed array satisfies the equality constraint exactly.

The rule has two parts: a mapping step, which sends an arbitrary index tuple to the canonical tuple of its permutation orbit, and a lookup step, which returns the stored value at that canonical tuple as the value for the original tuple. Correctness of the rule depends entirely on the guarantee that the equality constraint holds for the tensor being reconstructed, since the rule assumes without re-derivation that every tuple in an orbit shares a single value.

The Mapping Step

Sorting the Symmetrized Positions

Given an arbitrary index tuple, the mapping step extracts the indices occupying the symmetrized positions and rearranges them into non-decreasing order, while indices at any non-symmetrized positions retain their original placement. The resulting tuple is the canonical representative of the orbit to which the original tuple belongs.

Uniqueness of the Canonical Tuple

Every orbit under permutation of the symmetrized positions contains exactly one non-decreasing arrangement, so the mapping step always produces a single, well-defined canonical tuple regardless of which member of the orbit was supplied as input. This uniqueness is what allows the lookup step to proceed without ambiguity.

The Lookup Step

Reading the Stored Value

Once the canonical tuple is determined, the reconstruction rule retrieves the value that was stored for that tuple in the independent selection and assigns it as the value of the original, arbitrary tuple:

where i denotes the arbitrary index tuple and canonical(i) denotes the result of the mapping step applied to i.

Consistency With the Equality Constraint

Because the equality constraint requires all tuples in an orbit to share one value, the lookup step never encounters a conflict between the value implied by the original tuple and the value stored at the canonical tuple; the rule simply exposes the value already guaranteed to be shared.

Applying the Rule Across the Full Tensor

Iterating Over All Index Tuples

To reconstruct the complete component array from the independent selection, the reconstruction rule is applied once for every index tuple in the full index range, each time performing the mapping step followed by the lookup step, until every entry of the full array has been assigned a value.

Complexity of Reconstruction

The mapping step for a single tuple requires only a sort of the symmetrized indices, an operation whose cost depends on the number of symmetrized positions rather than on the dimension of the underlying vector space, making the reconstruction rule efficient to apply even when the full component array is large.

Role Within the Symmetric Tensor Framework

Inverse of Independent Selection

The reconstruction rule acts as the operational inverse of independent selection: where selection reduces the full array to a minimal set of representatives, reconstruction expands that minimal set back into the full array, and applying selection to a reconstructed array returns exactly the original independent selection.

Dependence on Correct Symmetrization

The reconstruction rule presumes that the tensor being described is genuinely symmetric in the declared positions; if a tensor only approximately satisfies the equality constraint, applying the reconstruction rule strictly would overwrite any small discrepancies between orbit members with the single stored representative value, effectively enforcing exact symmetry on the output.