symmetric algebra
The symmetric algebra is the tensor algebra after you let factors commute — it turns abstract vectors into ordinary commuting variables. Once order no longer matters, products of vectors behave exactly like monomials, and the whole structure becomes a coordinate-free polynomial ring built on a vector space. It is the free commutative algebra on the space.
Given an R-module V, the symmetric algebra S(V) is the quotient of the tensor algebra T(V) by the two-sided ideal generated by all v ⊗ w − w ⊗ v. The induced product is commutative, and S(V) = ⊕_{k≥0} S^k(V) is graded, with S^k(V) the symmetric k-tensors — the symmetric analogue of Λ^k(V). It satisfies the universal property: any R-linear map V -> A into a commutative R-algebra A extends uniquely to an algebra homomorphism S(V) -> A.
If V is free of rank n with basis x_1, ..., x_n, then S(V) ≅ R[x_1, ..., x_n], the polynomial ring in n variables, and dim S^k(V) = C(n + k − 1, k) counts degree-k monomials. Thus the symmetric algebra is the intrinsic, basis-free incarnation of polynomials, and its graded pieces house the symmetric multilinear forms.
For dim V = 2 with basis x, y: S^2(V) has basis x^2, xy, y^2 (dim 3 = C(3, 2)), and S(V) ≅ R[x, y]. Compare Λ^2(V), which is only 1-dimensional, spanned by x ∧ y.
The symmetric square is spanned by monomials; the exterior square by a single wedge.
Over a field of characteristic 0 there is a clean splitting V ⊗ V ≅ S^2(V) ⊕ Λ^2(V), with dim S^2 = n(n+1)/2 and dim Λ^2 = n(n−1)/2 summing to n^2. In characteristic 2 this decomposition fails, and symmetric and alternating tensors no longer cleanly split.