15.18.2 Tensor Symmetry Component Equality Check

Tensor Symmetry Component Equality Check is the precise, minimal-effort formulation of the direct definitional test for tensor symmetry, specifying exactly which pairs of components must be compared, how redundant comparisons can be eliminated by grouping indices into equivalence classes, and how the resulting comparison count can be made explicit for a tensor of given order and dimension.

Formulating the Minimal Comparison Set

Equivalence Classes of Index Tuples

An index tuple of length d, each entry ranging over the n basis directions, determines an orbit under the action of the symmetric group permuting the positions of the tuple: two tuples lie in the same orbit precisely when one is a rearrangement of the other. The Component Equality Check is founded on the observation that verifying symmetry requires comparing components only within each such orbit, since the Component Constraint asserts equality precisely among the components indexed by tuples in a common orbit, and imposes no relation between tuples belonging to different orbits.

Reduction to Sorted Representatives

Every orbit of index tuples has a unique representative given by sorting its entries into non-decreasing order; the Component Equality Check therefore proceeds by grouping all index tuples according to their sorted representative and, within each group, confirming that every tuple in the group has the same component value as the group's designated representative. This reformulation replaces an unstructured pairwise comparison with a partition of all index tuples into disjoint groups, each to be checked internally for constant value.

Counting the Required Comparisons

Number of Orbits

The number of distinct orbits, equivalently the number of sorted representative tuples, equals the number of ways to choose a multiset of size d from n symbols, given by the binomial coefficient of n plus d minus one, choose d; this is exactly the dimension of the space of symmetric tensors, confirming that each orbit corresponds to precisely one independent component that the Component Constraint permits to vary freely.

Size of Each Orbit

The number of tuples within a given orbit is determined by the multiplicities of repeated entries in its sorted representative, via the multinomial coefficient

where m_1 through m_n are the multiplicities with which each of the n symbols appears in the tuple, summing to d. An orbit with all distinct entries has the maximal size of d factorial, while an orbit in which a single symbol is repeated d times (corresponding to a pure diagonal component) has size exactly one, requiring no comparison at all.

Total Comparison Cost

Summing, over all orbits, one fewer than the orbit size (since checking constancy within a group of size m requires m minus one comparisons against the chosen representative) gives the exact total number of equality comparisons needed for a complete Component Equality Check, a count strictly smaller than the naive n raised to the power d comparisons implied by checking every ordered tuple against every permutation of itself independently.

Algorithmic Formulation

Procedure Outline

The check proceeds by iterating once over all ordered index tuples, computing for each tuple its sorted representative, and recording the tensor's component value in a lookup structure keyed by that representative; whenever a second tuple maps to a representative already recorded, its component value is compared against the stored value, and any mismatch immediately certifies that the tensor is not symmetric. This procedure visits each of the n raised to the power d ordered tuples exactly once, achieving the same asymptotic cost as the naive enumeration while performing genuinely fewer numerical comparisons, and while making explicit, through the lookup structure, exactly which single value each symmetric component is supposed to take.

Early Termination

Because a single mismatch is sufficient to conclude that the Component Constraint fails, the procedure can terminate as soon as the first discrepancy is found, making the Component Equality Check, in the common case of a genuinely non-symmetric tensor, typically far faster in practice than its worst-case comparison count would suggest.

Relation to the Broader Verification Procedure

Basis for the Symmetrization Residual Shortcut

The grouping of index tuples into orbits used here is exactly the grouping over which the symmetrization operator averages when computing the symmetric part of a general tensor, so that the Component Equality Check can be seen as the explicit, index-level unpacking of what the symmetrization residual test, described in the general Tensor Symmetry Verification Procedure, computes implicitly through a single linear operator application.

Use in Exact and Symbolic Settings

Because it relies only on equality comparisons rather than norms or thresholds, the Component Equality Check is the appropriate verification method whenever tensor components are exact, symbolic, or rational, such as in algebraic constructions built directly from apolarity computations or from explicit symmetric decompositions, situations in which the tolerance-based residual test described in the general Verification Procedure would be unnecessarily imprecise.