15.1.3 Tensor Symmetric Product Scope

Tensor Symmetric Product Scope is the delineation of which properties of the symmetric product operation $\odot$ itself, namely its commutativity, associativity, and bilinearity, together with its role as the multiplication making the graded pieces into a single algebra, are treated within this branch, as distinct from computational expansion algorithms and additional-structure-dependent operations left outside scope.

What Falls Within Scope

Commutativity of the Symmetric Product

Within scope is the basic fact that the symmetric product of two vectors does not depend on their order,

a direct consequence of the symmetrization operator averaging over every permutation, including the transposition swapping the two factors.

Associativity of the Symmetric Product

Also within scope is the fact that the symmetric product extends unambiguously to any number of factors regardless of how they are grouped,

both sides equal to the fully symmetrized product $u\odot v\odot w$, so parentheses may be omitted entirely when writing a symmetric product of several vectors.

Bilinearity

Within scope as well is the bilinearity of the symmetric product in each argument, $(\alpha u+\beta v)\odot w=\alpha (u\odot w)+\beta (v\odot w)$, established directly from linearity of the symmetrization operator and bilinearity of the ordinary tensor product it acts upon.

The Multiplication of the Symmetric Algebra

Extending the symmetric product from vectors to elements of any two graded pieces, ${\text{Sym}}^j\left(V\right)\times {\text{Sym}}^k\left(V\right)\to {\text{Sym}}^{j+k}\left(V\right)$, and confirming this extension is what makes the symmetric algebra a commutative associative algebra, falls within scope as the direct bridge between the pairwise product and the algebra structure noted at the level of structural scope.

What Falls Outside Scope

Explicit Expansion Algorithms

The combinatorial bookkeeping required to expand a symmetric product of many vectors into an explicit sum over all $k!$ permutations, or the more efficient algorithms used in practice to avoid this factorial blowup by working directly with multiset representatives, are outside scope; only the definition of the operation and its basic algebraic laws are addressed here, not its computational implementation.

Contraction With a Symmetric Product

Operations that contract a symmetric tensor against a covector or against another tensor, producing a tensor of lower degree, are outside this scope, since contraction requires pairing an upper index against a lower index and is a distinct operation from the symmetric product itself, which only combines tensors of the same variance into a tensor of higher degree.

Inner-Product-Dependent Structure

If $V$ is equipped with an inner product, the symmetric powers inherit an induced inner product under which distinct basis symmetric products are not, in general, orthonormal without an explicit normalizing factor; this additional structure, and the correction factors it requires, is outside the present scope, which treats the symmetric product purely algebraically, independent of any choice of inner product on $V$.

Relation Between the Included Properties

Commutativity and Associativity Together Give a Well-Defined Multi-Factor Product

Commutativity alone would permit reordering only two factors at a time; it is the combination of commutativity and associativity, both within scope, that guarantees a symmetric product of arbitrarily many vectors in any order and any grouping produces the same, unambiguous result, matching the coordinate-free definition of a symmetric tensor as invariant under every permutation, not merely under transpositions.