4.21.5 Tensor Symmetric Algebra Preparation

Tensor Symmetric Algebra Preparation is the body of facts about symmetric multilinear maps that must be in place before the symmetric algebra can be constructed and verified to have its expected properties: the universal property specific to symmetric maps, the identification of the defining ideal to quotient by, and the assembly of the individual symmetric powers into a single commutative graded algebra.

What Must Be Assembled First

The Symmetric Universal Property

Before the symmetric algebra is built, the correct universal property for symmetric maps must be isolated: for every symmetric multilinear map f: V × ... × V → W (n copies), there must exist a unique linear map f̃: Symⁿ(V) → W with f̃(v₁⋯vₙ) = f(v₁,...,vₙ). This is the symmetric analogue of the tensor product's universal property, guaranteeing that Symⁿ(V), however it is constructed, is the correct universal recipient for symmetric n-linear maps.

The Ideal Generated by Commutator-Type Elements

Constructing Sym(V) = ⊕ₙ Symⁿ(V) as a quotient of the tensor algebra T(V) = ⊕ₙ V^{⊗n} requires identifying the correct two-sided ideal J to quotient by: the ideal generated by all elements v ⊗ w - w ⊗ v for v, w ∈ V. Verifying that quotienting by this specific ideal, rather than the ideal I used for the exterior algebra, produces a quotient where the symmetric component pattern holds for the induced symmetric product is a preparatory step resting entirely on the symmetric multilinear pattern already established.

Deriving Commutativity of the Product From the Pattern

From the Generating Relation to Full Commutativity

Working inside T(V)/J, the relation v ⊗ w ≡ w ⊗ v for degree-one elements, imposed directly by the generators of J, is exactly the symmetric slot-exchange rule at degree two. Because the ideal J is two-sided, this relation propagates through products with other elements, so the induced symmetric product ⋯ on T(V)/J satisfies v ⋅ w = w ⋅ v for all v, w ∈ V, matching the symmetric pattern already required of multilinear maps of arity two.

Extending to Full Commutativity of the Algebra

Unlike the exterior algebra, where swapping factors introduces a sign depending on their degrees, the symmetric algebra is commutative outright at every degree: α ⋅ β = β ⋅ α for all α, β ∈ Sym(V), with no sign correction of any kind, since the generating relation itself carries no sign; this is a direct structural consequence of the symmetric pattern's lack of any transposition-induced sign change.

Assembling the Graded Algebra Structure

Direct Sum Across All Degrees

The symmetric algebra Sym(V) = ⊕_{n≥0} Symⁿ(V) collects every symmetric power into a single graded vector space, with the symmetric product respecting the grading: Symᵖ(V) ⋅ Symᑫ(V) ⊆ Sym^{p+q}(V). Unlike the exterior algebra, Symⁿ(V) is nonzero for every n ≥ 0 even when n exceeds dim(V), since repeated arguments are permitted rather than forbidden.

Associativity Inherited From the Tensor Algebra

Associativity of the symmetric product follows directly from associativity of the tensor product in T(V), since the symmetric product is defined as the tensor product followed by projection to the quotient T(V)/J, and this projection is a ring homomorphism; no independent verification specific to the symmetric structure is required once J is confirmed to be a two-sided ideal.

Dimension Count as a Consistency Check

Generating Function for Dimensions

The preparatory dimension count, dim(Symⁿ(V)) = C(d+n-1, n) for V of dimension d, is confirmed using the basis of non-decreasing-index symmetric products established from the symmetric basis component pattern. Summing across all degrees using generating functions gives

more precisely ∑ₙ C(d+n-1,n) tⁿ = 1/(1-t)^d, matching the generating function for the Hilbert series of the polynomial ring F[x₁,...,x_d], a check that the symmetric power construction has produced the expected sequence of dimensions.

Confirming the Universal Property Once the Algebra Is Built

Verifying Existence of the Factorization

Once Symⁿ(V) is constructed as T^n(V)/J_n, existence of the factorization f̃ for a given symmetric f follows the same reasoning used for the tensor product's universal property: f, being multilinear, factors through V^{⊗n} as f̂, and since f is symmetric, f̂ vanishes on the ideal J_n, so f̂ descends to a well-defined linear map f̃ on the quotient Symⁿ(V).

Verifying Uniqueness of the Factorization

Uniqueness follows because the symmetric products v₁⋯vₙ span Symⁿ(V), so any two linear maps agreeing with f on these spanning products must agree everywhere on Symⁿ(V), completing the preparatory work needed to confirm Symⁿ(V) genuinely satisfies the universal property for symmetric multilinear maps of arity n.

Identification With the Polynomial Ring

Matching Generators and Relations

Once the graded commutative algebra Sym(V) is assembled, comparing its generators, one for each basis vector of V, and its only relation, commutativity of multiplication, with the generators and relations of the polynomial ring F[x₁,...,x_d] confirms the isomorphism Sym(V) ≅ F[x₁,...,x_d], completing the preparation by connecting the abstract quotient construction to the concrete and familiar algebra of polynomials.

Why This Preparation Precedes the Symmetric Algebra Proper

Separating the Combinatorial Pattern From the Algebraic Construction

Establishing the symmetric pattern, invariance under any argument permutation, the non-decreasing-tuple component structure, prior to constructing the symmetric algebra keeps the combinatorial content of symmetry separate from the algebraic task of building a quotient ring with well-defined commutative multiplication; the symmetric algebra construction then transports already-established facts about symmetric multilinear maps into statements about a concrete graded commutative algebra, mirroring step for step the analogous preparation carried out for the exterior algebra, but without any sign corrections at any stage.