15.7.5 Tensor Symmetrization Result Space

Tensor Symmetrization Result Space is the image of the symmetrization operator Sym, that is, the set of all tensors that can arise as the output Sym(S) for some input tensor S of the given rank, coinciding exactly with the subspace of totally symmetric tensors satisfying the equality constraint on all n index positions. This space is a proper linear subspace of the full rank-n tensor space whenever n is at least two and the underlying vector space has dimension at least two, since not every rank-n tensor is symmetric, and it inherits a vector space structure from the ambient tensor space because Sym is linear.

Identifying the result space precisely, rather than merely describing the operator that produces it, matters because many properties of interest, such as dimension counts, basis choices, and compatibility with other tensor operations, are properties of this space itself rather than properties of the operator's formula; the result space is the object of ultimate interest, and Sym is the tool used to reach it.

Characterizing the Result Space

Equivalence With the Symmetric Subspace

The result space of Sym equals the set of tensors satisfying the total symmetry component constraint: every symmetric tensor T is its own image under Sym, since Sym fixes tensors already satisfying the constraint, and every image Sym(S) is symmetric, since the permutation sum underlying Sym always produces a symmetric output. These two facts together establish the set-theoretic equality between the result space and the symmetric subspace.

Closure Under Linear Combination

Because the result space coincides with the solution set of a homogeneous linear system, namely the equality constraint's defining equations, it is automatically closed under addition and scalar multiplication: a linear combination of tensors each satisfying the equality constraint also satisfies it, confirming directly that the result space is a genuine linear subspace of the full tensor space.

Dimension of the Result Space

Counting via Orbit Structure

The dimension of the result space equals the number of independent components available to a totally symmetric rank-n tensor over a d-dimensional vector space, which is the number of permutation orbits of index tuples, given by the multiset-counting formula:

matching exactly the independent component count derived earlier from the equality constraint and from independent selection.

Comparison With the Ambient Space Dimension

This dimension is generally much smaller than the dimension d^n of the full, unconstrained tensor space once n exceeds one, and the ratio between the two dimensions shrinks as n grows, reflecting how strongly the total symmetry constraint reduces the space of admissible tensors relative to the space of all possible rank-n tensors.

Basis for the Result Space

Symmetrized Basis Tensors

A natural spanning set for the result space is obtained by applying Sym to the standard basis tensors of the full tensor space, namely the tensors with a single component equal to one at a chosen index tuple and zero elsewhere; symmetrizing each such basis tensor produces a symmetric tensor supported on the entire orbit of that index tuple.

Selecting a Linearly Independent Subset

Restricting this construction to basis tuples in canonical, non-decreasing order yields a linearly independent set whose size matches the dimension of the result space exactly, since distinct canonical tuples produce symmetrized tensors supported on disjoint orbits and therefore cannot be linear combinations of one another; this set forms an explicit basis for the result space.

Interaction With Other Constructions

Result Space as the Domain of Independent Selection

Every tensor in the result space, by virtue of satisfying the equality constraint, admits an independent selection and a reconstruction rule as described for symmetric tensors generally; the result space is precisely the domain on which those constructions are valid and well-defined, since they presuppose the equality constraint that defines this space.

Relation to the Symmetric Bilinear and Multilinear Form Structures

Every tensor in the result space, when interpreted through the symmetric multilinear form structure, defines a totally symmetric multilinear map on n vector arguments; conversely, every totally symmetric multilinear map arises from some tensor in this result space, so the result space of Sym stands in exact correspondence with the space of totally symmetric multilinear forms of the same rank, unifying the component-level and functional-level descriptions of symmetric tensors within a single subspace.