15.9.5 Tensor Symmetric Power Polynomial Relation

Tensor Symmetric Power Polynomial Relation is the precise correspondence, under the isomorphism between the symmetric algebra Sym(V) and a polynomial ring, between a symmetric power v^{odot n} of a vector v and the n-th power of the linear polynomial expression associated with v, establishing that raising a vector to a symmetric power on the tensor side matches exactly raising a linear form to an ordinary power on the polynomial side. This relation is the sharpest and most computationally direct instance of the general correspondence between symmetric tensors and homogeneous polynomials, since it involves no summation over multiple terms but a single, pure power of a single linear expression.

The relation matters because it turns questions about symmetric powers into questions about ordinary algebra familiar from elementary polynomial manipulation, such as the binomial theorem, allowing results proven in the polynomial setting to be transported directly back into statements about tensors, and vice versa, without needing to re-derive them from the tensor definitions each time.

Establishing the Correspondence

The Linear Form Associated With a Vector

Given a basis e_1 through e_d of V with dual coordinate functions x_1 through x_d, a vector v with coordinates v^1 through v^d corresponds, under the polynomial isomorphism, to the linear polynomial expression:

with the coefficients of the linear form given directly by the coordinates of v.

The Power Correspondence

The symmetric power polynomial relation states that the associated homogeneous polynomial of v^{odot n} is exactly the n-th ordinary power of this linear form:

so that the tensor operation of taking a symmetric power translates exactly into the polynomial operation of raising the corresponding linear expression to a power.

Expansion via the Binomial and Multinomial Theorems

Two-Dimensional Case and the Binomial Theorem

When the underlying vector space has dimension two, with coordinates x_1 and x_2, the linear form ax_1 plus bx_2 raised to the n-th power expands via the binomial theorem into a sum over k from 0 to n of the binomial coefficient n choose k, times a to the (n-k), times b to the k, times x_1 to the (n-k), times x_2 to the k; each term corresponds to one basis element of Sym^n(V), confirming that the coefficients appearing in the expansion of v^{odot n} match exactly the binomial coefficients from elementary algebra.

General Dimension and the Multinomial Theorem

For a linear form in d variables raised to the n-th power, the multinomial theorem generalizes the binomial expansion, producing a sum over all ways of distributing n total factors among the d variables, with each term weighted by the corresponding multinomial coefficient; this expansion directly reproduces the coordinates of the symmetric power v^{odot n} with respect to the basis of symmetric powers of the individual basis vectors e_1 through e_d.

Consequences for Computing With Symmetric Powers

Coefficients as Multinomial Counts

Because the polynomial relation identifies the coefficients of v^{odot n}, expressed in the natural basis of symmetric power monomials, with multinomial coefficients, computing the explicit component values of a symmetric power reduces to a combinatorial counting problem already fully solved by elementary combinatorics, rather than requiring a fresh computation of the underlying permutation sum from the symmetrization operator.

Verification via Small Cases

For n equal to two, the relation predicts that v^{odot 2} corresponds to the polynomial ℓ(x) squared, whose expansion contains a doubled cross term for each pair of distinct variables, matching directly the earlier observation that the off-diagonal components of a rank-two symmetric tensor appear with a factor of two in the quadratic form expansion, confirming the polynomial relation against a case already established independently.

Broader Significance Within the Symmetric Power Structure

Anchoring the Structure in Familiar Algebra

The polynomial relation is what makes the symmetric power structure immediately tractable using tools no more advanced than ordinary polynomial expansion, and it is the specific mechanism by which the abstract algebra role of the symmetric product, described generally as an isomorphism to a polynomial ring, becomes concrete and computable in the particular case of powers of a single vector.

Foundation for General Symmetric Tensor Computation

Since every element of Sym^n(V) is a linear combination of symmetric powers of basis vectors, and every symmetric power of a general vector expands via the multinomial theorem into a combination of these basis powers, the polynomial relation for symmetric powers provides, in combination, a complete computational bridge between arbitrary symmetric tensors and arbitrary homogeneous polynomials of the corresponding degree.