15.19.2 Tensor Symmetric Product Notation

Tensor Symmetric Product Notation is the notational convention for the operation that combines several vectors into a single symmetric tensor by symmetrized multiplication, commonly written using a small circle between the vectors, and it is the operation-level counterpart to the Symmetric Power Notation, which names the space in which the results of this operation live.

The Basic Operation

Defining the Symmetric Product of Vectors

Given vectors v_1 through v_k in a vector space V, their symmetric product, written v_1 circle v_2 circle ... circle v_k, is defined as the symmetrization of their ordinary tensor product:

where the sum runs over all permutations sigma of the k vectors and the plain circled-cross symbol denotes the ordinary, unsymmetrized tensor product. The result lands in S^k V, and this circled-dot symbol is the standard Symmetric Product Notation used to denote it.

Recovering the Pure Power Form

When all k vectors coincide, equal to a single vector v, the symmetric product collapses to the pure power form used throughout Tensor Symmetric Decomposition Structure, since every permutation of k identical vectors leaves the tensor product unchanged and the averaging sum reduces to k factorial identical copies divided by k factorial:

confirming that the Symmetric Product Notation, applied to repeated arguments, is fully compatible with the tensor-power notation for pure power forms.

Algebraic Properties Encoded by the Notation

Commutativity

Because the symmetric product is defined as an average over all orderings, it is manifestly commutative: reordering the vectors v_1 through v_k in any way leaves the symmetric product unchanged, which is precisely why the operation is written using a symbol, the circle, that suggests an ordinary commutative multiplication rather than the ordered tensor product symbol.

Multilinearity

The symmetric product is linear in each of its arguments separately, since both the tensor product and the averaging sum used to define it are linear operations; this multilinearity is what allows the symmetric product of a linear combination of vectors to be expanded term by term, exactly as ordinary multiplication distributes over addition, and it is this expansion that produces the mixed cross terms appearing in the Tensor Quadratic Form Component Expression when a symmetric product of two vector sums is written out explicitly.

Associativity Within the Symmetric Algebra

Within the symmetric algebra structure carried by the direct sum of all symmetric powers, the symmetric product is associative, so that longer chains of symmetric products can be parenthesized in any order without changing the result, matching the associativity of ordinary polynomial multiplication under the identification of S(V) with a polynomial ring.

Notation for General Elements of a Symmetric Power

Spanning Sets Built from Symmetric Products

Every element of S^d V can be written as a linear combination of symmetric products of d basis vectors of V, and choosing one representative symmetric product for each multiset of d basis indices (rather than for each ordered tuple) gives a basis for S^d V, directly realizing the dimension count of S^d V as the number of such multisets, consistent with the binomial coefficient formula discussed under the Symmetric Power Notation.

Relation to Monomial Notation

Under the identification of S^d V with homogeneous polynomials, the symmetric product of d basis covectors corresponds exactly to the ordinary monomial formed by multiplying the corresponding coordinate variables, so that the Symmetric Product Notation and standard polynomial monomial notation describe the same basis elements from two different but fully compatible notational traditions.

Use in Expressing Decompositions and Polarization

Symmetric Decompositions in Product Notation

A symmetric decomposition, elsewhere written as a sum of pure power forms using tensor-power notation, can equally be written using the Symmetric Product Notation applied to d copies of each decomposition vector, and the two notations are used interchangeably depending on whether the emphasis is on the tensor-algebraic origin of the terms or on their multiplicative, polynomial-like combination.

Compatibility with Polarization

The Tensor Quadratic Form Polarization Relation, which recovers a symmetric bilinear tensor from its associated quadratic form, can be restated using the Symmetric Product Notation as recovering the coefficient of the symmetric product x circle y from combinations of the diagonal values Q(x), Q(y), and Q(x + y), making explicit that polarization is fundamentally the operation of extracting mixed symmetric-product coefficients from a purely diagonal (single-argument) evaluation.