7.19.3 Tensor Independent Component Symmetry Reduction

Tensor Independent Component Symmetry Reduction is the general reduction in the number of independent components that results when a tensor fulfills the Tensor Component Symmetric Tensor Role across all of its indices simultaneously, extending the pairwise Tensor Component Symmetric Reduction to tensors of arbitrary order by counting the number of ways to select index values with repetition allowed, rather than counting entries within a single triangle of a two index table.

From Pairwise Reduction to General Reduction

The Limitation of the Pairwise Count

The Tensor Component Symmetric Reduction, as originally described, applies to a single designated pair of indices and produces a count of n times the quantity n plus one, divided by two, for that pair alone. This figure does not directly extend to a tensor whose every index, not merely one pair, participates in a fully symmetric relationship, since higher order symmetry links together far more of the tensor's positions than a single pair can capture.

The General Formula for Full Symmetry

For a tensor of order r fulfilling the Tensor Component Symmetric Tensor Role across all r of its indices, defined on a space of dimension n, the number of independent components is given by the multiset coefficient:

This expression counts the number of ways to choose r index values, allowing repetition and disregarding order, from the n available values in the Tensor Component Index Range, since any independent component of a fully symmetric tensor of order r is uniquely identified by a multiset of r index values, arranged in a single fixed, non-decreasing order.

Why Non-Decreasing Combinations Suffice

Every Ordering of the Same Values Is Equal

Because every pair of indices in a fully symmetric tensor follows the Tensor Component Symmetric Equality Rule, any two components whose index lists are permutations of one another must share exactly the same value. Selecting the single non-decreasing arrangement as the representative for each multiset of index values therefore captures exactly one independent component per multiset, with every other ordering of the same values being a derived value equal to that representative.

Repetition Is Permitted, Unlike the Antisymmetric Case

Unlike the fully antisymmetric case, where repeated index values force a component to vanish, a fully symmetric tensor places no such restriction on repeated values, since the equality rule imposes no constraint when the indices being exchanged already coincide. This is why the counting formula allows repetition, in contrast to the strictly distinct values required for the corresponding antisymmetric count.

Illustration

All three orderings of the multiset containing the values one, one, and two share exactly the same component value, and the non-decreasing arrangement is chosen as their single independent representative.

Consistency With the Pairwise Case

Recovering the Original Formula at Order Two

Setting r equal to two in the general multiset coefficient formula gives the quantity n plus one choose two, which simplifies to n times the quantity n plus one, divided by two, exactly matching the count originally established by the pairwise Tensor Component Symmetric Reduction. The general formula therefore extends, rather than replaces, the pairwise result.

No Upper Bound From the Dimension

Unlike the antisymmetric case, where the order cannot exceed the dimension without forcing every component to vanish, a fully symmetric tensor can have an order greater than the dimension of the underlying space and still possess a positive number of independent components, since repetition of index values is permitted without restriction.

Relationship to Other Tensor Concepts

Tensor Independent Component Symmetry Reduction generalizes the Tensor Component Symmetric Reduction from a single index pair to the full order of a tensor fulfilling the Tensor Component Symmetric Tensor Role, drawing on the same underlying Tensor Component Symmetric Equality Rule that governs the pairwise case. It forms an essential part of the Tensor Independent Component Structure for any tensor exhibiting complete symmetry across its indices, standing as the direct counterpart to the Tensor Independent Component Antisymmetry Reduction.