15.8.2 Tensor Symmetric Product Permutation Averaging

Tensor Symmetric Product Permutation Averaging averages permutations to compute symmetric tensor products, key in invariant theory and multilinear algebra.

Tensor Symmetric Product Permutation Averaging is the specific mechanism, internal to the definition of the symmetric product, by which the ordinary tensor product of several factors is converted into a totally symmetric tensor: every one of the ways the factors could have been arranged is summed and then divided by the total number of arrangements, so that the result treats all orderings of the original factors as equally contributing to the outcome. This averaging is exactly the symmetrization operator applied after the ordinary tensor product has already combined the factors, and it is what guarantees that the symmetric product T odot R does not privilege the order in which T and R, or their constituent vectors, happened to be written down.

Examining this averaging step specifically, apart from the general symmetrization operator it invokes, clarifies how the symmetric product's defining properties, commutativity and associativity, follow directly from the mechanics of averaging over permutations rather than being separate assumptions imposed on the product.

The Averaging Mechanism in the Product Definition

Expanding the Product Before Averaging

For vectors u_1 through u_n, the ordinary tensor product u_1 tensor ... tensor u_n has components equal to the product of the corresponding coordinates, u_1^{i1} times u_2^{i2} times ... times u_n^{in}, arranged according to the fixed order in which the vectors were listed; this ordered array generally fails the equality constraint, since permuting the index positions corresponds to permuting which vector's coordinate appears in which slot.

Averaging Over Every Arrangement

The permutation averaging step then replaces this ordered array with the average, over all n factorial permutations σ of the argument order, of the same product of coordinates rearranged according to σ:

(u_{1} ⊙ \dots ⊙ u_{n})_{i_{1} \dots i_{n}} = \frac{1}{n!} \sum_{σ}^{} u_{σ (1)}^{i_{1}} \dots u_{σ (n)}^{i_{n}}

which is exactly the symmetrization operator's permutation sum, with normalization, applied to the components of the ordinary tensor product.

Commutativity as a Consequence of Averaging Over All Arrangements

Why Order of Factors Does Not Matter

Because the averaging step sums over every one of the n factorial orderings of the original factors, starting from any particular ordering of u_1 through u_n produces exactly the same collection of n factorial terms as starting from a differently ordered list of the same vectors; the sum, and hence the average, is therefore identical no matter which order the factors were originally written in, which is precisely the commutativity property of the symmetric product.

Explicit Two-Factor Illustration

For two vectors u and v, the permutation average has only two terms, corresponding to the identity permutation and the single transposition, giving u odot v equal to one half times u tensor v plus v tensor u; visibly, this expression is unchanged if u and v are swapped, since swapping merely exchanges the two terms in the sum, matching the general commutativity argument in the smallest nontrivial case.

Extending Averaging to Products of Symmetric Tensors

Averaging Applied After Combining Higher-Rank Factors

When the factors being combined are themselves symmetric tensors T of rank p and R of rank q rather than plain vectors, the same permutation averaging step is applied to the p plus q indices of the ordinary tensor product T tensor R, summing over every rearrangement of those combined indices and dividing by (p plus q) factorial.

Interaction With the Factors' Existing Symmetry

Because T and R are already individually symmetric under permutations within their own p and q indices respectively, many of the (p plus q) factorial terms in the full permutation sum coincide with each other before averaging even begins, though the averaging formula still divides by the full (p plus q) factorial rather than by a reduced count, since the formula is defined uniformly regardless of any partial symmetry already present in the inputs.

Role of Averaging in Guaranteeing Associativity

Nested Averaging Reduces to Single-Step Averaging

Forming (u odot v) odot w by first averaging u and v, then combining with w and averaging again over the relevant permutations, produces the same result as averaging u, v, and w together in one three-factor permutation sum, because the two-step averaging process, when fully expanded, revisits every one of the six orderings of the three original vectors exactly once, matching the single-step three-factor average term for term.

Consistency Across Grouping Choices

This term-for-term matching is what guarantees the symmetric product's associativity, confirming that permutation averaging, applied consistently regardless of how the factors are grouped, always converges to the same fully symmetrized combination of the original inputs.