Symmetric and Exterior Algebras

From the tensor algebra, we construct quotient algebras by imposing symmetry or antisymmetry relations. These yield symmetric and exterior algebras, fundamental in geometry and physics.

DefinitionSymmetric Tensor Product

The symmetric tensor product $V \odot W$ is the quotient of $V \otimes W$ by the relation: $\mathbf{v} \otimes \mathbf{w} \sim \mathbf{w} \otimes \mathbf{v}$

Elements of $\text{Sym}^k(V)$ (the $k$ -fold symmetric product) are symmetric tensors, invariant under permutations.

Example: $\text{Sym}^2(\mathbb{R}^n)$ consists of symmetric $n \times n$ matrices (dimension $\binom{n+1}{2}$ ).

DefinitionExterior (Wedge) Product

The exterior product $\Lambda^k(V)$ is the quotient of $V^{\otimes k}$ by antisymmetry: $\mathbf{v}_1 \wedge \mathbf{v}_2 \wedge \cdots \wedge \mathbf{v}_k$

satisfies $\mathbf{v}_{\sigma(1)} \wedge \cdots \wedge \mathbf{v}_{\sigma(k)} = \text{sgn}(\sigma) \mathbf{v}_1 \wedge \cdots \wedge \mathbf{v}_k$ for any permutation $\sigma$ .

Key property: $\mathbf{v} \wedge \mathbf{v} = 0$ , so $\mathbf{v} \wedge \mathbf{w} = -\mathbf{w} \wedge \mathbf{v}$ .

Dimension: $\dim(\Lambda^k(V)) = \binom{n}{k}$ where $n = \dim(V)$ .

ExampleExterior Algebra in $\mathbb{R}^3$

For $V = \mathbb{R}^3$ with basis $\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$ :

$\Lambda^1(V) = V$ (dimension 3)
$\Lambda^2(V)$ has basis $\{\mathbf{e}_1 \wedge \mathbf{e}_2, \mathbf{e}_1 \wedge \mathbf{e}_3, \mathbf{e}_2 \wedge \mathbf{e}_3\}$ (dimension 3)
$\Lambda^3(V)$ has basis $\{\mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3\}$ (dimension 1)

The wedge product $\mathbf{e}_i \wedge \mathbf{e}_j$ generalizes the cross product.

TheoremDeterminants and Exterior Algebra

For $n$ -dimensional space $V$ , $\Lambda^n(V)$ is one-dimensional. The wedge product of $n$ vectors: $\mathbf{v}_1 \wedge \mathbf{v}_2 \wedge \cdots \wedge \mathbf{v}_n = \det([\mathbf{v}_1 \cdots \mathbf{v}_n]) \cdot (\mathbf{e}_1 \wedge \cdots \wedge \mathbf{e}_n)$

The determinant emerges as the coefficient in this unique top form.

DefinitionDifferential Forms

In differential geometry, differential forms are sections of exterior bundles. A $k$ -form on manifold $M$ assigns to each point an element of $\Lambda^k(T_p^*M)$ (the exterior power of the cotangent space).

Integration and Stokes' theorem are formulated using differential forms, providing coordinate-free calculus on manifolds.

Remark

The exterior algebra underlies modern differential geometry and topology. Wedge products encode oriented volumes: $\mathbf{v} \wedge \mathbf{w}$ represents the oriented parallelogram. In physics, electromagnetic field strength is a 2-form, unifying electric and magnetic fields in relativistic formulation. The antisymmetry captures the essence of alternating multilinear algebra.