15.2.1 Tensor Symmetric Structure Area

The Tensor Symmetric Structure Area explores symmetric properties of tensors, foundational in algebra for understanding multilinear structures and their invariants.

Tensor Symmetric Structure Area is the branch of application in which the graded ring structure of the symmetric algebra, rather than any individual symmetric tensor, is the object of direct use, underlying the construction of polynomial rings and projective space in algebraic geometry, the ring of invariants in invariant theory, and the free-algebra characterization used to define polynomial functors.

Algebraic Geometry

Affine Space as the Spectrum of a Symmetric Algebra

The coordinate ring of affine space $K^{n}$ is, by the identification of the symmetric algebra with a polynomial ring, exactly $Sym (V^{*})$ for $V$ an $n$ -dimensional vector space, so that affine space itself is recovered from the symmetric algebra structure as its associated spectrum, giving the symmetric algebra a foundational role in the construction of the most basic algebraic varieties.

Projective Space and the Veronese Embedding

Projective space is built from the graded pieces ${Sym}^{k} (V^{*})$ of the symmetric algebra taken together, and the Veronese embedding, which re-embeds projective space using only the degree- $d$ symmetric tensors as coordinates, is defined directly in terms of the graded structure of $Sym (V^{*})$ , sending a point to the tuple of values of every degree- $d$ monomial at that point.

Invariant Theory

The Ring of Invariants as a Subring

When a group $G$ acts on $V$ , it acts correspondingly on each symmetric power ${Sym}^{k} (V)$ , and classical invariant theory studies the subring of elements of the symmetric algebra fixed by this action,

{(Sym (V))}^{G} = ⨁_{k} {({Sym}^{k} (V))}^{G}

which is finitely generated as an algebra whenever $G$ is a reductive group, a foundational finiteness theorem that relies essentially on the ring structure of the symmetric algebra rather than on any single symmetric tensor in isolation.

Classical Examples

The elementary symmetric polynomials arising when $G$ is the full symmetric group acting by permuting coordinates, and the discriminant and resultant of a polynomial arising from more elaborate group actions, are both concrete instances of invariants living inside the ring structure of a symmetric algebra, connecting this structural area directly back to the symmetrization area's treatment of elementary symmetric polynomials.

Polynomial Functors

The Free Commutative Algebra Characterization in Use

The universal property of the symmetric algebra, that $Sym (V)$ is the free commutative algebra generated by $V$ , is used directly to define the symmetric power as a functor: any linear map $f : V \to W$ induces a unique algebra homomorphism $Sym (f) : Sym (V) \to Sym (W)$ extending it, and this functoriality, following directly from the universal property, is what makes ${Sym}^{k}$ a well-behaved polynomial functor usable throughout representation theory and algebraic geometry.

Base Change

The universal property also governs how the symmetric algebra interacts with extension of scalars: for a field extension $K \subseteq L$ , the symmetric algebra of $V \otimes_{K} L$ over $L$ is canonically identified with $Sym (V) \otimes_{K} L$ , a compatibility that follows from the free-algebra universal property holding simultaneously over both fields.

Summary Contrast With the Other Application Areas

Structure Versus Individual Tensors

Where the form area and the areas involving mechanics and statistics treat a single symmetric tensor, such as a metric or a covariance matrix, as the object of interest, the structure area treats the entire graded collection $⨁_{k} {Sym}^{k} (V)$ and its ring and functorial properties as the object of interest, making this area the natural point of contact between symmetric tensors and commutative algebra, algebraic geometry, and category theory.