15.8 Tensor Symmetric Product Operation

Tensor Symmetric Product Operation is the construction that combines two or more symmetric tensors, or two or more individual vectors, into a new symmetric tensor of higher rank, defined by forming the ordinary tensor product of the inputs and then applying the symmetrization operator to project the result onto the totally symmetric subspace. Denoted with a symbol such as an odot or a juxtaposition without an explicit tensor product sign, the symmetric product ensures that the output automatically satisfies the equality constraint on all of its indices, regardless of whether the ordinary tensor product of the inputs already did.

This operation is the primary means by which symmetric tensors of higher rank are built up from simpler pieces, particularly from vectors, and it plays a role for symmetric tensors analogous to the role the wedge product plays for antisymmetric tensors: just as the wedge product of vectors produces a totally antisymmetric tensor, the symmetric product of vectors produces a totally symmetric tensor, with the key structural difference being the absence of any sign changes under argument exchange.

Definition of the Operation

Symmetric Product of Vectors

For vectors u_1 through u_n in a vector space V, the symmetric product is defined as the symmetrization of their ordinary tensor product:

where the right-hand tensor product symbol denotes the ordinary, unsymmetrized tensor product, and Sym is the symmetrization operator described earlier.

Symmetric Product of Symmetric Tensors

More generally, given two symmetric tensors T of rank p and R of rank q, their symmetric product T odot R is defined by forming the ordinary tensor product T tensor R, of rank p plus q, and then applying the symmetrization operator across all p plus q indices, producing a totally symmetric tensor of that combined rank.

Basic Properties

Commutativity

The symmetric product of two symmetric tensors is commutative, T odot R equals R odot T, since reordering the factors before symmetrizing merely reorders the terms inside the resulting permutation sum, leaving the fully symmetrized total unchanged; this stands in direct contrast to the ordinary tensor product, which is generally not commutative.

Associativity

The symmetric product is associative, so (T odot R) odot Q equals T odot (R odot Q), because both expressions reduce, after full expansion, to the symmetrization of the ordinary tensor product T tensor R tensor Q across all indices simultaneously; this allows the symmetric product of several factors to be written without parentheses.

Bilinearity

The symmetric product is linear in each factor separately, inheriting this property from the linearity of the ordinary tensor product and the linearity of the symmetrization operator, so scaling or adding one factor scales or distributes correspondingly across the resulting symmetric product.

Relation to Coefficients and the Multinomial Structure

Symmetric Product of Repeated Vectors

Taking the symmetric product of a single vector v with itself n times produces a tensor closely related to the n-th power appearing in the associated homogeneous polynomial: v odot v odot ... odot v, with n factors, evaluated through the bilinear or multilinear form correspondence, reproduces the degree-n monomial in the coordinates of v up to an overall combinatorial factor tied to the multinomial coefficients governing how many ordered arrangements collapse into each symmetric term.

Building the Independent Component Basis

The basis vectors for the result space of the symmetrization operator, discussed as the symmetrized standard basis tensors, are themselves symmetric products of the standard basis vectors e_1 through e_d taken with appropriate repetition, so every symmetric tensor can be expressed as a linear combination of symmetric products of basis vectors, mirroring how every ordinary tensor is a linear combination of ordinary tensor products of basis vectors.

The Symmetric Product Within the Algebraic Structure

Symmetric Algebra as the Home of the Operation

Collecting the symmetric products of all ranks, starting from rank zero, scalars, through rank one, vectors themselves, and upward, forms a graded algebra called the symmetric algebra on V, with the symmetric product serving as the multiplication operation of this algebra; the commutativity and associativity properties established above are exactly the axioms required for this multiplication to make the symmetric algebra a commutative, associative algebra.

Contrast With the Ordinary Tensor Algebra

The ordinary tensor algebra, built from the unsymmetrized tensor product, is associative but not commutative; the symmetric algebra can be realized as a quotient of the tensor algebra by the two-sided ideal generated by differences u tensor v minus v tensor u, and the symmetric product operation described here is precisely the induced multiplication on that quotient.