15.7.2 Tensor Symmetrization Normalization Factor

The Tensor Symmetrization Normalization Factor ensures symmetric tensor components are properly scaled, crucial for consistent algebraic operations in tensor algebra.

Tensor Symmetrization Normalization Factor is the scalar coefficient, equal to the reciprocal of n factorial for a rank-n symmetrization, that multiplies the sum over all permutations of a tensor's indices in order to make the symmetrization operator a true projector rather than merely an averaging-like map that scales its output. Without this factor, summing a tensor's components over every permutation of a set of n index positions produces a result that is n factorial times too large whenever the tensor is already symmetric, since each of the n factorial identical terms would be added rather than averaged; the normalization factor corrects this by dividing the sum back down to the scale of a single representative term.

The role of the normalization factor is not merely cosmetic. It is precisely what guarantees the defining property of a projector, namely that applying the operator twice produces the same result as applying it once, and it is what makes the symmetrization operator's output coincide exactly with the original tensor whenever that tensor already satisfies the symmetric component constraint, rather than merely producing a rescaled copy of it.

Definition Within the Symmetrization Operator

The Unnormalized Sum

For a rank-n tensor S with components S_{i1...in}, the sum over all permutations of the index positions is:

\sum_{σ}^{} S_{i_{σ (1)} \dots i_{σ (n)}}

where σ ranges over all n factorial permutations of the positions 1 through n.

Insertion of the Normalization Factor

The symmetrization operator applies the normalization factor of 1 divided by n factorial to this sum:

Sym (S)_{i_{1} \dots i_{n}} = \frac{1}{n!} \sum_{σ}^{} S_{i_{σ (1)} \dots i_{σ (n)}}

turning the raw permutation sum into an average over the n factorial permutations.

Why the Factor Must Equal One Over n Factorial

Idempotence Requirement

The essential requirement placed on the symmetrization operator is idempotence, Sym(Sym(S)) equals Sym(S) for every tensor S. Applying the unnormalized permutation sum a second time to an already-permutation-summed tensor multiplies the result by an additional factor of n factorial, since Sym(S) is already invariant under permutation and summing an invariant quantity over all n factorial permutations simply adds n factorial identical copies. Dividing by n factorial exactly once cancels this extra factor and restores idempotence.

Fixed-Tensor Requirement

A second requirement is that Sym act as the identity on any tensor T that already satisfies the symmetric component constraint. Summing the components of such a T over all n factorial permutations yields n factorial copies of the single value T_{i1...in}, since every permuted index tuple already equals the original by the constraint; dividing this sum by n factorial recovers T_{i1...in} exactly, confirming that the normalization factor is what makes Sym fix already-symmetric tensors rather than scale them upward.

Consequences of Omitting or Miscalculating the Factor

Effect of Omission

If the normalization factor were omitted entirely, the resulting operator would map every tensor, symmetric or not, to a rescaled version of its symmetric part, off by a factor of n factorial; the image of this unnormalized operator would still be the subspace of symmetric tensors, but the operator would fail to be idempotent and would fail to act as the identity on tensors already lying in that subspace.

Rank Dependence of the Factor's Magnitude

Because the factor is 1 divided by n factorial, its magnitude shrinks rapidly as the rank n increases, reflecting the rapidly growing number of terms, n factorial, being averaged together; for rank 2 the factor is one half, for rank 3 it is one sixth, and for rank 4 it is one twenty-fourth, a pattern that continues for every subsequent rank.

Extension to Partial Symmetrization

Restricting the Sum to a Subgroup of Permutations

When only a subset of a tensor's index positions is to be symmetrized, the permutation sum runs only over permutations of that subset, holding the remaining positions fixed, and the normalization factor is correspondingly 1 divided by k factorial, where k is the number of positions being symmetrized rather than the full rank n.

Consistency With Total Symmetrization

Setting k equal to n in this partial symmetrization recovers the full symmetrization operator and its associated normalization factor of 1 divided by n factorial, confirming that partial symmetrization with its own factor is a strict generalization of, and reduces correctly to, the fully symmetric case already described.