15.8.3 Tensor Symmetric Product Commutative Behavior

Tensor Symmetric Product Commutative Behavior is the property that the symmetric product of two symmetric tensors does not depend on the order in which the two factors are written, so that T odot R equals R odot T for every pair of symmetric tensors T and R over the same underlying vector space. This behavior stands in direct contrast to the ordinary tensor product, under which T tensor R and R tensor T are generally distinct tensors, and it is a direct structural consequence of the permutation averaging step built into the definition of the symmetric product, rather than an additional assumption imposed separately.

Commutative behavior is one of the two algebraic properties, together with associativity, that make the symmetric product suitable as the multiplication operation of a commutative algebra, and it is what allows expressions built from several symmetric products to be freely rearranged without changing their value, a convenience unavailable when working with the ordinary, non-commutative tensor product.

Statement and Proof of Commutativity

The Commutativity Identity

For symmetric tensors T of rank p and R of rank q, commutative behavior is the identity:

with both sides understood as symmetric tensors of rank p plus q, obtained respectively by symmetrizing T tensor R and R tensor T.

Proof via Reordering the Permutation Sum

Both T tensor R and R tensor T, when their p plus q indices are summed over every permutation as part of the symmetrization step, produce sums that range over exactly the same set of (p plus q) factorial rearranged component values, since any rearrangement reachable starting from the ordering used in T tensor R is also reachable starting from the ordering used in R tensor T, merely via a different, correspondingly relabeled permutation; the two permutation sums therefore total to the same value, and after applying the identical normalization factor, T odot R and R odot T coincide exactly.

Illustration With Vector Factors

Two-Vector Case

For two vectors u and v, both of rank one, the symmetric product reduces to the explicit average u odot v equals one half times (u tensor v plus v tensor u); swapping u and v in this formula merely swaps the two summands, leaving the sum, and hence u odot v, completely unchanged, giving the simplest direct verification of commutative behavior.

Extension to Several Vector Factors

For n vectors u_1 through u_n, commutative behavior extends beyond swapping just two factors to swapping any two, or rearranging all n, since the underlying permutation averaging step already sums over every one of the n factorial orderings of the vectors; consequently u_1 odot u_2 odot ... odot u_n is unaffected by any reordering of the list u_1 through u_n, not merely by pairwise swaps.

Commutativity and the Failure of Commutativity in the Ordinary Product

Ordinary Tensor Product Is Not Commutative

The ordinary tensor product T tensor R generally differs from R tensor T as soon as either factor has rank at least one, because the component arrays T tensor R and R tensor T place the indices of T and R in different relative positions; only after symmetrization does this positional distinction get averaged away, which is exactly why the ordinary tensor product itself lacks commutative behavior while the symmetric product built from it possesses it.

Symmetrization as the Source of the Commutativity

Commutative behavior of the symmetric product is therefore not an independent property to be verified separately from the definition of symmetrization; it follows directly and automatically once the permutation averaging step is understood to sum over the entire symmetric group of rearrangements, since that full-group averaging is precisely what erases any dependence on the original relative ordering of the factors.

Role of Commutativity Within the Symmetric Algebra

Compatibility With Associativity

Commutative behavior combines with the associativity of the symmetric product to allow any finite collection of symmetric tensors combined via repeated symmetric products to be reordered and regrouped arbitrarily without affecting the result, matching the two axioms required for the symmetric product to serve as the multiplication of a commutative, associative algebra built on the graded pieces of symmetric tensors of every rank.

Contrast With the Exterior Product

Commutative behavior of the symmetric product stands opposite to the behavior of the wedge product used to build antisymmetric tensors, where swapping two factors introduces a sign change, u wedge v equals negative v wedge u; the symmetric product's unconditional commutativity, with no sign change under any reordering, is the structural feature that most sharply distinguishes it from its antisymmetric counterpart.