15.4.4 Tensor Symmetric Component Independent Selection

Tensor Symmetric Component Independent Selection is the procedure of choosing, from the full array of components of a symmetric tensor, a specific subset that contains exactly one component from each permutation orbit and thereby forms a minimal, non-redundant description of the tensor. Because the symmetric component equality constraint forces every index tuple within an orbit to carry the same value, a single selected tuple per orbit is sufficient to determine the entire tensor: any component not directly selected can be recovered by permuting a selected tuple's indices until they match the requested tuple.

This selection is distinct from redundancy reduction in emphasis: redundancy reduction focuses on the storage and computational benefit of avoiding duplication, while independent selection focuses on identifying which specific set of index tuples constitutes a valid, non-overlapping, and complete set of degrees of freedom for the tensor. A selection is independent when no two chosen tuples belong to the same orbit, and it is complete when every orbit is represented by at least one chosen tuple.

Defining Property of a Valid Selection

Non-Overlap Condition

A selection of index tuples is independent precisely when no permutation of the symmetrized indices maps one selected tuple onto another selected tuple. If two distinct selected tuples were related by such a permutation, they would be forced by the equality constraint to carry the same value, making one of them a dependent rather than an independent degree of freedom.

Coverage Condition

A selection is complete when every possible index tuple in the tensor's full index range can be reached from some selected tuple by a permutation of the symmetrized positions. Together, non-overlap and coverage mean the selection contains exactly one representative from each orbit, matching the orbit count exactly.

Standard Selection Rule

Non-Decreasing Index Tuples

The most common independent selection rule chooses, from each orbit, the unique tuple whose indices are arranged in non-decreasing order. Since every orbit under the full permutation group contains exactly one non-decreasing arrangement, this rule automatically satisfies both the non-overlap and coverage conditions without requiring an auxiliary check.

Enumeration of the Selected Set

The selected set under this rule corresponds bijectively to the multisets of size n drawn from d index values, and its size is given by the same count that governs the number of independent components:

The tuples in this selected set can be enumerated in lexicographic order, giving a deterministic indexing scheme that assigns a unique position to each independent component.

Using the Selection to Reconstruct the Tensor

Mapping an Arbitrary Tuple to Its Selected Representative

Given any index tuple, its corresponding selected representative is obtained by sorting the symmetrized index positions into non-decreasing order while leaving any non-symmetrized positions untouched. The value at the original tuple is then read directly from the value stored at the sorted representative, since the equality constraint guarantees the two are equal.

Building the Tensor From the Selection Alone

A symmetric tensor can be fully specified by supplying values only for the independent selection, together with the rule that any other component equals the value at its orbit's representative. This gives a compact and unambiguous way to define or communicate a symmetric tensor without listing its full, redundant component array.

Distinction From Related Notions

Selection Versus Redundancy Reduction

Independent selection identifies which tuples are kept; redundancy reduction is the practical use of that identification to save storage and computation. The two are complementary: a valid selection is a prerequisite for a correct redundancy reduction scheme, since reduction without a well-defined, non-overlapping, complete selection would either lose information or retain unnecessary duplication.

Selection Under Partial Symmetry

When a tensor is symmetric only in a subset of its indices, independent selection sorts only the indices within that subset when forming the canonical representative, leaving indices outside the symmetric subset free to vary independently across the selected set, so the count of independent components combines the orbit count of the symmetric subset with the unrestricted range of the remaining indices.