15.18.3 Tensor Symmetry Permutation Check

Tensor Symmetry Permutation Check is the full, definitional verification of tensor symmetry against every permutation of the index positions individually, serving as the baseline formulation of the Component Constraint from which the more efficient Component Equality Check and Slot Exchange Check are derived as provably equivalent shortcuts.

The Full Definitional Statement

Quantifying Over the Entire Symmetric Group

Given a tensor T of order d, the Permutation Check declares T symmetric precisely when, for every permutation sigma belonging to the symmetric group on d letters, and for every assignment of index values i_1 through i_d, the following identity holds:

This is the literal, unabridged rendering of the Component Constraint, quantifying explicitly over all d factorial elements of the symmetric group, and it is the definition against which every other, more efficient verification procedure must be justified.

Distinguishing the Definition from Its Verification

It is important to separate the Permutation Check as a definition, stating precisely what symmetry means, from the Permutation Check as an algorithm, which would naively test every one of the d factorial permutations directly. The definition is unconditionally correct and requires no justification beyond stating what "symmetric" means; the naive algorithmic approach to testing it, however, is computationally wasteful, and this is precisely the inefficiency that motivates the reduced procedures.

Why the Full Check Is Rarely Executed Directly

Combinatorial Growth of the Permutation Group

The number of permutations to test, d factorial, grows extremely quickly with the order d of the tensor: while manageable for order two or three, it becomes prohibitive for tensors of even moderate order, since factorial growth quickly outpaces any polynomial-time verification budget. Executing the Permutation Check literally, by testing every permutation independently for every index assignment, is therefore reserved, in practice, for tensors of very low order or for symbolic verification of small illustrative examples.

Redundancy Among the Permutations Tested

A large part of the inefficiency of the literal Permutation Check lies in redundancy: verifying invariance under a given permutation sigma, once invariance has already been confirmed under every permutation that generates sigma as a product, adds no new information, since the invariance under sigma follows automatically by composing the already-verified invariances. This redundancy is exactly what the Slot Exchange Check exploits by restricting attention to a generating set of only d minus one transpositions, and what the Component Equality Check exploits by grouping permutations according to the orbit of the index tuple they act upon rather than testing them one at a time.

The Permutation Check as a Correctness Standard

Role in Justifying Reduced Procedures

Although rarely executed in its literal form, the Permutation Check remains essential as the standard of correctness: any proposed shortcut, such as checking only adjacent transpositions or only orbit representatives, must be proven logically equivalent to the full Permutation Check before it can be trusted, and the proofs given under the Slot Exchange Check and the Component Equality Check are precisely such equivalence arguments, showing that their reduced sets of conditions imply, via group generation or orbit membership, the full condition stated here.

Role in Theoretical Arguments

Beyond practical verification, the Permutation Check formulation is the one used directly in theoretical arguments about symmetric tensors, such as the proof of Transformation Preservation, where an arbitrary permutation sigma is considered directly, and in the representation-theoretic framing of Subspace Invariance, where the fixed points of the entire symmetric group action, not merely of a generating subset, are the object under study; in these theoretical contexts, working with the full permutation group, rather than a reduced generating set, is often the more natural and transparent approach, even though it would be inefficient as a computational procedure.

Structuring the Full Check for Partial Efficiency

Restricting to a Minimal Generating Set Without Losing Rigor

A practical compromise retains the logical structure of the full Permutation Check, quantifying conceptually over the entire symmetric group, while executing only the reduced set of tests known to be logically equivalent, whether via adjacent transpositions or via orbit representatives; this compromise is what is actually implemented whenever tensor symmetry is verified computationally, with the full Permutation Check serving as the specification being satisfied rather than as the literal sequence of operations performed.

Consistency Across All Formulations

Because the Permutation Check, the Slot Exchange Check, and the Component Equality Check are all provably equivalent characterizations of the same Component Constraint, any one of them can be used interchangeably as the operational definition of tensor symmetry, with the choice among them determined solely by which is most convenient for the task at hand, whether that task is a hand proof, a symbolic computation, or a large-scale numerical verification.