15.9.3 Tensor Symmetric Power Basis Construction

Tensor Symmetric Power Basis Construction is the explicit procedure for building a basis of the degree-n graded piece Sym^n(V) using symmetric products of powers of the individual basis vectors of V, providing a concrete, enumerable set of tensors whose linear span is the entire space of rank-n symmetric tensors. Given a basis e_1 through e_d of the underlying vector space V, this construction produces one basis element of Sym^n(V) for every way of distributing n total factors of repetition among the d basis directions, matching exactly the dimension given by the symmetric power dimension relation.

The construction is significant because it turns the abstract existence of a basis, guaranteed by general linear algebra, into an explicit, canonical, and easily indexed family of tensors, each one directly expressible as a symmetric product of powers of basis vectors; this explicit basis is what makes concrete computation with symmetric tensors, such as expressing an arbitrary symmetric tensor in coordinates, practically possible.

The Construction Procedure

Indexing by Exponent Tuples

For each choice of non-negative integers k_1 through k_d satisfying k_1 plus k_2 plus ... plus k_d equals n, the construction assigns the basis element:

formed by taking the k_1-th symmetric power of e_1, the k_2-th symmetric power of e_2, and so on, then combining all d results with the symmetric product; because the symmetric product is commutative and associative, this expression is well-defined without needing to specify an order for the combination.

Correspondence With Exponent Tuples

Each distinct tuple (k_1, ..., k_d) of non-negative integers summing to n produces one basis element, and this correspondence is exactly the same combinatorial bijection between such tuples and multisets of size n drawn from d values that underlies the dimension formula, giving a direct match between the count of constructed basis elements and the required dimension of Sym^n(V).

Linear Independence of the Constructed Set

Distinct Monomials Under the Polynomial Isomorphism

Under the isomorphism between Sym(V) and the polynomial ring in d variables, the constructed basis element indexed by (k_1, ..., k_d) corresponds exactly to the monomial x_1^{k_1} x_2^{k_2} ... x_d^{k_d}; since distinct exponent tuples produce distinct monomials, and distinct monomials are linearly independent as polynomials, the corresponding symmetric power combinations are likewise linearly independent as tensors.

Spanning the Full Graded Piece

Because the count of these linearly independent elements equals the dimension of Sym^n(V) exactly, the constructed set is not merely independent but a complete basis, spanning the entire degree-n graded piece; no additional symmetric tensor of rank n lies outside the span of this explicit family of symmetric power combinations.

Relation to Independent Component Selection

Basis Elements as Canonical Representatives

The exponent tuple (k_1, ..., k_d) indexing a given basis element corresponds directly to the canonical, non-decreasing index tuple used in independent component selection for symmetric tensors, with k_j recording how many times the basis direction j appears among the n repeated indices; this connects the explicit power-based basis construction directly to the earlier, more general discussion of selecting one representative component per permutation orbit.

Coordinates of a General Symmetric Tensor in This Basis

Expressing an arbitrary symmetric tensor T in this basis assigns to each exponent tuple (k_1, ..., k_d) the coefficient equal to T's component at the corresponding canonical index tuple, scaled by the appropriate multinomial factor arising from the polynomial relation for symmetric powers, giving an explicit formula for translating between the component description of T and its representation as a combination of symmetric power basis elements.

Practical Use of the Constructed Basis

Computation via Multinomial Coefficients

Because each basis element corresponds to a pure monomial, computations involving symmetric tensors expressed in this basis, such as evaluating the associated homogeneous polynomial or computing a symmetric product of two tensors given in this basis, reduce to ordinary polynomial arithmetic, leveraging the multinomial coefficients that appear when expanding products of monomials.

Foundation for Extending to Symmetric Tensors of Mixed Rank Combinations

The same basis construction extends naturally to describe bases for symmetric products of general symmetric tensors, not only vectors, by first expressing each factor in its own power-based basis and then combining the resulting expressions using the bilinearity of the symmetric product, making the power-based basis construction the practical starting point for essentially all concrete computation within the symmetric algebra.