Chapter 100. Symmetric Algebra

Symmetric algebra studies commutative multilinear structure. It provides the algebraic framework for polynomials, symmetric tensors, and higher-order homogeneous forms.

Where exterior algebra encodes antisymmetry and orientation, symmetric algebra encodes commutativity and repetition.

The tensor product distinguishes order:

u \otimes v \neq v \otimes u.

The symmetric algebra identifies these expressions:

u \otimes v = v \otimes u.

This construction produces the algebra of polynomial-like expressions generated by a vector space.

Symmetric algebra appears throughout algebraic geometry, representation theory, differential equations, optimization, and physics.

100.1 Motivation

Suppose $V$ is a vector space over a field $F$ .

We often wish to form expressions such as

x^2, \qquad xy, \qquad x^3y,

where multiplication is commutative:

xy = yx.

Ordinary tensor products do not enforce commutativity.

For example,

x \otimes y

and

y \otimes x

are generally distinct tensors.

The symmetric algebra modifies tensor algebra by imposing the relations

u\otimes v = v\otimes u.

The resulting structure behaves like polynomial multiplication.

100.2 Tensor Algebra Review

The tensor algebra of $V$ is

T(V) = \bigoplus_{k=0}^{\infty} V^{\otimes k}.

Multiplication is tensor product concatenation:

(u_1\otimes \cdots \otimes u_p) \otimes (v_1\otimes \cdots \otimes v_q) = u_1\otimes \cdots \otimes u_p \otimes v_1\otimes \cdots \otimes v_q.

Tensor algebra is associative but not commutative.

The symmetric algebra is obtained by forcing commutativity.

100.3 Definition of Symmetric Algebra

The symmetric algebra of $V$ is denoted

S(V).

It is defined as the quotient

S(V) = T(V)/I,

where $I$ is the ideal generated by elements

u\otimes v - v\otimes u.

Thus tensors differing only by permutation become equal.

The image of

u_1\otimes\cdots\otimes u_k

in $S(V)$ is written

u_1u_2\cdots u_k.

Multiplication therefore becomes commutative:

uv = vu.

The symmetric algebra behaves like a polynomial algebra generated by vectors.

100.4 Symmetric Tensors

A tensor is symmetric if permuting indices leaves it unchanged.

For example,

u\otimes v + v\otimes u

is symmetric.

More generally,

T(v_{\sigma(1)},\ldots,v_{\sigma(k)}) = T(v_1,\ldots,v_k)

for every permutation $\sigma$ .

The space of symmetric $k$ -tensors is denoted

\operatorname{Sym}^k(V).

This space corresponds to homogeneous polynomials of degree $k$ .

100.5 Symmetrization

Any tensor can be converted into a symmetric tensor by averaging over permutations.

v_1\otimes\cdots\otimes v_k

is a tensor, its symmetrization is

\frac{1}{k!} \sum_{\sigma\in S_k} v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(k)}.

This operation projects tensors onto the symmetric subspace.

Exterior algebra used antisymmetrization. Symmetric algebra instead uses averaging without signs.

100.6 Polynomial Interpretation

Suppose

V = \operatorname{span}\{x_1,\ldots,x_n\}.

Then

S(V) \cong F[x_1,\ldots,x_n],

the polynomial ring in $n$ variables.

Under this correspondence:

Symmetric algebra element	Polynomial
$x_i$	Linear variable
$x_ix_j$	Degree-2 monomial
$x_i^k$	Power monomial
Linear combinations	Polynomials

Thus symmetric algebra generalizes polynomial algebra to arbitrary vector spaces.

100.7 Grading

The symmetric algebra is graded:

S(V) = \bigoplus_{k=0}^{\infty} S^k(V).

Here:

Space	Meaning
$S^0(V)$	Scalars
$S^1(V)$	Vectors
$S^2(V)$	Quadratic tensors
$S^3(V)$	Cubic tensors

Multiplication satisfies

S^p(V)\cdot S^q(V) \subseteq S^{p+q}(V).

Thus polynomial degrees add under multiplication.

100.8 Basis of Symmetric Powers

Suppose

\{e_1,\ldots,e_n\}

is a basis for $V$ .

Basis elements for $S^k(V)$ are monomials

e_1^{a_1}\cdots e_n^{a_n}

with

a_1+\cdots+a_n=k.

The dimension equals the number of monomials of degree $k$ in $n$ variables:

\dim(S^k(V)) = \binom{n+k-1}{k}.

This combinatorial formula is fundamental in representation theory and algebraic geometry.

100.9 Quadratic Forms

Symmetric tensors naturally represent quadratic forms.

A quadratic form is an expression

Q(x) = x^TAx,

where $A$ is symmetric:

A=A^T.

The associated bilinear form is

B(u,v) = u^TAv.

Quadratic forms correspond to symmetric second-order tensors.

Examples include:

Quadratic form	Interpretation
$x^2+y^2$	Euclidean norm
$x^2-y^2$	Hyperbolic form
$ax^2+bxy+cy^2$	Conic sections

Symmetric algebra therefore underlies much of classical geometry.

100.10 Homogeneous Polynomials

A polynomial is homogeneous of degree $k$ if every monomial has degree $k$ .

Examples:

Polynomial	Degree
$x^2+xy+y^2$	2
$x^3+y^3$	3
$xyz$	3

Homogeneous polynomials correspond precisely to symmetric tensors in $S^k(V)$ .

This relationship is central in algebraic geometry.

100.11 Universal Property

The symmetric algebra satisfies a universal property.

Let

A

be a commutative algebra and let

f : V \to A

be linear.

Then there exists a unique algebra homomorphism

\widetilde{f} : S(V) \to A

extending $f$ .

Thus symmetric algebra is the free commutative algebra generated by $V$ .

This property characterizes symmetric algebra abstractly.

100.12 Comparison with Exterior Algebra

Symmetric algebra and exterior algebra arise from opposite symmetry conditions.

Structure	Relation imposed
Symmetric algebra	$u\otimes v = v\otimes u$
Exterior algebra	$u\otimes v = -v\otimes u$

Consequences:

Algebra	Behavior
Symmetric	Repetition allowed
Exterior	Repetition vanishes

For example:

These two constructions dominate multilinear algebra.

100.13 Symmetric Bilinear Forms

A bilinear form

B : V\times V \to F

is symmetric if

B(u,v)=B(v,u).

Examples include inner products and quadratic forms.

Symmetric bilinear forms correspond naturally to symmetric tensors of degree two.

Matrix representations satisfy

A=A^T.

Such matrices play central roles in spectral theory and geometry.

100.14 Symmetric Powers of Linear Maps

Suppose

T : V \to W

is linear.

Then $T$ induces maps

S^k(T) : S^k(V) \to S^k(W).

The action is defined by

S^k(T)(v_1\cdots v_k) = T(v_1)\cdots T(v_k).

Thus symmetric powers form functorial constructions.

These constructions are important in representation theory.

100.15 Symmetric Algebra in Geometry

Symmetric algebra appears naturally in geometry.

Examples include:

Area	Role
Algebraic geometry	Polynomial coordinate rings
Differential geometry	Symmetric tensors
Riemannian geometry	Metric tensors
Optimization	Hessian matrices
Classical mechanics	Energy functions

Polynomial equations defining geometric objects are elements of symmetric algebras.

100.16 Metric Tensors

A metric tensor is a symmetric bilinear form

g : V\times V \to F.

In coordinates,

g(u,v) = \sum_{i,j} g_{ij}u_iv_j.

The coefficients satisfy

g_{ij}=g_{ji}.

Metrics define lengths, angles, and distances.

In differential geometry and relativity, the metric tensor is one of the central objects of study.

100.17 Hessians

The Hessian matrix of a smooth function records second derivatives:

H_f = \left( \frac{\partial^2 f} {\partial x_i\partial x_j} \right).

Mixed partial derivatives satisfy

\frac{\partial^2 f} {\partial x_i\partial x_j} = \frac{\partial^2 f} {\partial x_j\partial x_i}

under suitable regularity assumptions.

Therefore the Hessian is symmetric.

Symmetric algebra provides the natural framework for higher-order derivative structures.

100.18 Representation Theory

Symmetric powers generate important representations of groups.

For example, if

V=\mathbb{C}^n,

then

S^k(V)

is a representation of the general linear group

GL(n).

These symmetric-power representations appear in:

Field	Application
Representation theory	Irreducible modules
Algebraic geometry	Projective embeddings
Quantum theory	Bosonic states
Harmonic analysis	Polynomial representations

Symmetric tensors describe particles obeying Bose-Einstein statistics.

100.19 Example

Let

V=\operatorname{span}\{x,y\}.

Then:

Degree	Basis
$S^0(V)$	$1$
$S^1(V)$	$x,y$
$S^2(V)$	$x^2,xy,y^2$
$S^3(V)$	$x^3,x^2y,xy^2,y^3$

Notice:

xy=yx.

Thus

x\otimes y

and

y\otimes x

represent the same element in the symmetric algebra.

The dimension formula gives

\dim(S^2(V)) = \binom{2+2-1}{2} = 3.

Similarly,

\dim(S^3(V)) = \binom{2+3-1}{3} = 4.

100.20 Symmetric Algebra and Physics

In quantum mechanics, symmetric tensors describe bosons.

Bosonic wavefunctions remain unchanged under particle exchange:

\psi(x_1,x_2) = \psi(x_2,x_1).

This symmetry corresponds exactly to symmetric tensor structure.

Exterior algebra instead describes fermions, where exchange changes sign.

Thus symmetric and exterior algebras encode the two fundamental particle symmetries in quantum theory.

100.21 Summary

Symmetric algebra studies commutative multilinear structure.

Its defining property is:

uv = vu.

Key ideas include:

Concept	Meaning
Symmetric tensor	Permutation-invariant tensor
Symmetric algebra $S(V)$	Free commutative algebra on $V$
Symmetric power $S^k(V)$	Degree- $k$ homogeneous structure
Polynomial interpretation	Commutative monomials
Quadratic forms	Symmetric degree-2 tensors

Symmetric algebra provides the algebraic foundation for polynomials, metric tensors, Hessians, and commutative multilinear structures. Together with tensor algebra and exterior algebra, it forms one of the central frameworks of modern multilinear mathematics.