# Chapter 100. Symmetric Algebra

# 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.

If

$$
v_1\otimes\cdots\otimes v_k
$$

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:

| Expression | Symmetric algebra | Exterior algebra |
|---|---|
| \(u^2\) | Nonzero | Zero |
| Swapping factors | No sign change | Sign reversal |

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.