15.8.4 Tensor Symmetric Product Degree Addition

Tensor Symmetric Product Degree Addition is the rule governing the rank, or degree, of the tensor produced by a symmetric product operation, stating that the symmetric product of a rank-p symmetric tensor and a rank-q symmetric tensor is always a symmetric tensor of rank p plus q, with the ranks of the two factors adding together exactly as they would under the ordinary, unsymmetrized tensor product. This rule confirms that passing from the ordinary tensor product to the symmetric product, by inserting a permutation averaging step, changes which subspace the result lies in but does not change the fundamental bookkeeping of how many indices, or how much rank, the output carries.

The degree addition rule gives the symmetric product a graded structure: tensors of a fixed rank form a graded piece, and the symmetric product operation moves between these graded pieces in a way controlled entirely by ordinary addition of the input degrees. This grading is what allows the totality of symmetric tensors of every rank, together with the symmetric product, to be organized into a single algebraic object indexed by non-negative integers.

Statement of the Rule

Rank of the Symmetric Product

For a symmetric tensor T of rank p and a symmetric tensor R of rank q, both built over the same underlying vector space, the symmetric product satisfies:

matching the rank of the ordinary tensor product T tensor R that the symmetric product is built from before permutation averaging is applied.

Why Averaging Does Not Alter the Index Count

The permutation averaging step underlying the symmetric product rearranges the p plus q indices of the ordinary tensor product among themselves; it neither introduces new index positions nor removes existing ones, since every permutation appearing in the sum acts as a bijection on the same fixed set of p plus q positions, so the number of indices, and hence the rank, is preserved exactly through the averaging process.

Consequences for Repeated Symmetric Products

Rank of an n-Fold Symmetric Product of Vectors

Taking the symmetric product of n individual vectors, each of rank one, produces a tensor of rank n, since degree addition applied repeatedly gives one plus one, n times in succession, summing to n; this matches the direct definition of the symmetric product of n vectors as the symmetrization of their n-fold ordinary tensor product.

Rank Under Mixed Combinations

Combining, for instance, a rank-2 symmetric tensor with a rank-3 symmetric tensor via the symmetric product yields a rank-5 symmetric tensor, and this holds regardless of how the two original tensors were themselves built up, whether directly as arrays satisfying the equality constraint or as symmetric products of lower-rank pieces, since degree addition depends only on the ranks of the immediate factors being combined.

The Grading Induced on the Space of All Symmetric Tensors

Graded Pieces Indexed by Rank

Collecting the symmetric tensors of each rank n, starting from rank zero, into a single graded object, with rank n symmetric tensors forming the degree-n graded piece, the degree addition rule states precisely that the symmetric product operation sends the degree-p piece and the degree-q piece into the degree-(p+q) piece, which is the defining compatibility condition for a graded algebra multiplication.

Rank Zero and Rank One as Base Cases

The degree-zero piece consists of scalars, and the degree-one piece consists of the vectors of the underlying vector space themselves; the symmetric product of a scalar with any symmetric tensor simply rescales that tensor without changing its rank, consistent with rank zero adding to any rank n and leaving the sum equal to n, while the symmetric product of two vectors produces the rank-two symmetric tensors studied through the symmetric bilinear form structure.

Comparison With Degree Addition in Related Constructions

Consistency With the Ordinary Tensor Algebra Grading

The ordinary tensor algebra, built from the unsymmetrized tensor product, obeys the identical degree addition rule, since the ordinary tensor product of a rank-p and a rank-q tensor is always rank p plus q; the symmetric algebra inherits this same grading behavior because it arises as a quotient of the tensor algebra that respects rank, confirming that the symmetric product's degree addition rule is not a special feature unique to symmetrization but a property carried over faithfully from the underlying tensor product structure.

Parallel With Polynomial Degree

The degree addition rule for the symmetric product mirrors the ordinary rule for polynomial multiplication, where multiplying a degree-p polynomial by a degree-q polynomial produces a degree-(p+q) polynomial; this parallel is not coincidental, since the symmetric algebra on a finite-dimensional vector space is isomorphic to a polynomial ring, and the symmetric product corresponds under this isomorphism to ordinary polynomial multiplication, with rank corresponding exactly to polynomial degree.