15.7.1 Tensor Symmetrization Permutation Sum

Tensor Symmetrization Permutation Sum is the unnormalized sum, taken over every permutation of a chosen set of index positions, of the components of a tensor evaluated at each permuted index tuple, forming the raw computational core of the symmetrization operator before the normalization factor is applied. Where the normalization factor rescales the operator into a projector, the permutation sum is the combinatorial machinery that actually gathers together every rearrangement of the relevant indices, and it is the properties of this sum, rather than the constant scaling factor alone, that determine why the resulting symmetrized tensor satisfies the equality constraint.

Studying the permutation sum on its own, separate from the normalization that follows it, clarifies which structural features of symmetrization come from the act of summing over the permutation group and which come merely from the choice of scale; the sum by itself already guarantees permutation invariance of its output, and the normalization factor's only remaining job is to fix the overall size of that output.

Definition of the Sum

Summing Over the Full Symmetric Group

For a rank-n tensor S with components S_{i1...in}, the permutation sum over the positions 1 through n is:

where S_n denotes the symmetric group on n letters, containing n factorial elements, and each term in the sum is the original tensor evaluated at one particular permuted rearrangement of the requested index tuple.

Term Count and Independence From Choice of Labeling

The sum contains exactly n factorial terms for every fixed index tuple, one for each permutation σ, and this count is independent of the specific values of the indices i_1 through i_n, though repeated index values among the i_k can cause distinct permutations to produce identical terms.

Permutation Invariance of the Sum

Reindexing the Sum Under a Fixed Rearrangement

For any fixed permutation τ, evaluating P(S) at the τ-rearranged index tuple gives a sum over σ of S at the (στ)-rearranged tuple; since composition with τ is a bijection of the symmetric group onto itself, this sum ranges over exactly the same n factorial terms as the original sum, merely reordered, and therefore has the same total value.

Consequence: The Sum Is Always Symmetric

This reindexing argument shows that P(S) satisfies the symmetric component equality constraint for any input tensor S whatsoever, regardless of whether S itself was symmetric; the permutation sum alone, before any normalization, already produces an object obeying the equality constraint on the chosen index positions.

Behavior on Already-Symmetric Input

Collapsing to n Factorial Identical Terms

If the input tensor T already satisfies the equality constraint, every one of the n factorial terms in P(T) evaluated at a fixed tuple equals the same value T_{i1...in}, since permuting the indices of an already-symmetric tensor does not change its component value; the sum therefore collapses to n factorial times that single shared value.

Scale Discrepancy Corrected by Normalization

This n-factorial-fold scale-up is exactly the discrepancy that the normalization factor is designed to remove, confirming that the permutation sum handles the structural requirement of producing a permutation-invariant output, while the separate normalization factor handles only the requirement of matching scale, together giving the full symmetrization operator its dual properties of correct image and idempotence.

Relation to Partial and Restricted Sums

Sums Over a Subgroup

When symmetrization is intended over only a subset of index positions rather than all n, the permutation sum is restricted to range over permutations belonging to the subgroup that permutes only those chosen positions, leaving the remaining positions fixed in every term of the sum; the same reindexing argument applies within this subgroup, guaranteeing that the restricted sum is invariant under permutations of the chosen subset specifically.

Building Toward the Full Operator

The permutation sum, together with its associated normalization factor matched to the size of the permutation group being summed over, together constitute the symmetrization operator in full; the sum supplies the invariance property while the factor supplies the correct idempotent scale, and neither alone reproduces both defining properties of a true symmetrization projector.