15.9.1 Tensor Symmetric Power Repeated Factor

Tensor Symmetric Power Repeated Factor is the special case of the symmetric product in which a single vector, or a single symmetric tensor, is combined with itself a fixed number of times rather than with distinct other factors, producing what is called the n-th symmetric power of that vector or tensor. Denoted v^{odot n} for a vector v repeated n times, this construction is the direct tensor-algebra analogue of raising a number to a power, and it is the specific instance of the symmetric product from which the associated homogeneous polynomial of a symmetric tensor is most directly and transparently visible.

Because every factor in a symmetric power is identical, the permutation averaging step that defines the general symmetric product simplifies considerably: permuting the argument positions of n copies of the same vector cannot produce a different arrangement of distinct entries, so the averaging collapses in a way that reveals the tight relationship between symmetric powers and ordinary polynomial powers.

Definition of the Symmetric Power

The n-th Symmetric Power of a Vector

For a vector v and a positive integer n, the n-th symmetric power is defined as the n-fold symmetric product of v with itself:

a totally symmetric tensor of rank n, obtained by symmetrizing the ordinary tensor product of n copies of v.

Components of the Symmetric Power

Because every factor is the same vector v, the ordinary tensor product v tensor ... tensor v already has components equal to the product of coordinates v^{i1} v^{i2} ... v^{in}, which is itself invariant under any permutation of the index labels i_1 through i_n, since multiplication of numbers is commutative; consequently the permutation averaging step in the symmetrization leaves this component array entirely unchanged, and v^{odot n} has components exactly equal to v^{i1} v^{i2} ... v^{in} without any additional averaging effect.

Relation to the Associated Homogeneous Polynomial

Direct Correspondence to a Pure Power

Evaluating the multilinear form built from v^{odot n} on n copies of an arbitrary vector w, or more directly considering the associated homogeneous polynomial obtained by the quadratic-and-higher relation, reproduces the n-th power of the linear form defined by v, so the symmetric power v^{odot n} corresponds exactly to the monomial-like expression (linear form in v)^n, making it the tensor-algebra realization of raising a single linear quantity to a power.

Simplicity Relative to General Symmetric Products

Because a symmetric power involves only one distinct vector rather than several, its associated homogeneous polynomial factors completely into n identical linear factors, in contrast to the associated polynomial of a symmetric product of distinct vectors, which generally factors into n distinct linear factors or does not factor into linear pieces at all when the tensor is a sum of several such products.

Symmetric Power of a Higher-Rank Symmetric Tensor

Generalizing Beyond Vectors

The symmetric power construction extends beyond vectors to any symmetric tensor T of rank p, giving T^{odot n}, a symmetric tensor of rank np formed by taking the symmetric product of n copies of T with itself; by the degree addition rule, the rank multiplies the original rank p by the repetition count n.

Reduced Averaging Compared to Distinct Factors

As with the vector case, because all n factors in T^{odot n} are literally the same tensor T, the permutation averaging step acting on blocks of p indices at a time exhibits additional internal repetition beyond what appears when combining n genuinely distinct symmetric tensors, though the full averaging over all np indices is still required whenever the symmetric product’s general definition treats the combined index set as fully interchangeable rather than only block-interchangeable.

Symmetric Powers as Building Blocks

Spanning the Symmetric Subspace via Powers Alone

A notable structural fact is that the rank-n symmetric subspace over a vector space is spanned already by the symmetric powers v^{odot n} of individual vectors v alone, without needing symmetric products of distinct vectors as separate building blocks, since any symmetric product of distinct vectors can be recovered from a suitable linear combination of symmetric powers of sums of those vectors, via the same polarization identity that recovers a multilinear form from its diagonal evaluation.

Practical Use in Encoding Homogeneous Polynomials

Because a symmetric power directly encodes a pure power of a single linear form, symmetric powers are the natural tensor-algebra vehicle for representing individual monomial terms of a homogeneous polynomial, with general symmetric tensors, representing general homogeneous polynomials, built up as linear combinations of these symmetric power building blocks.