16.11 Tensor Exterior Product Relation

Tensor Exterior Product Relation is the set of defining algebraic relations governing the wedge product operation on alternating tensors, specifying how it combines elements of different degree, how it responds to reordering its factors, and how it reduces to zero on linearly dependent inputs, collectively fixing the exterior product uniquely up to the normalization convention chosen.

The Foundational Relation

The Anticommutation Relation for Vectors

At the most basic level, the exterior product relation for two vectors (or covectors) u and v states:

with the immediate consequence, obtained by setting u = v, that:

for every vector v. These two facts together are often taken as the generating relations from which the entire exterior algebra is constructed.

Bilinearity Relation

The exterior product also satisfies the distributive relations expected of any bilinear operation:

so the anticommutation relation and bilinearity together fully determine the product's behavior on any linear combination of vectors.

The Graded Relation at Higher Degree

Graded Anticommutativity

For elements α of degree p and β of degree q, the exterior product relation generalizes the simple vector anticommutation to a degree-dependent sign:

recovering the simple anticommutation relation exactly when p = q = 1.

Associativity Relation

The exterior product satisfies associativity, allowing multi-factor products to be written unambiguously without parentheses:

so the relation family, taken together, is what makes Λ•(V*) a well-defined associative graded algebra rather than merely a collection of pairwise-defined products.

The Vanishing Relation for Dependent Factors

Repeated Factor Vanishing

Directly extending v ∧ v = 0, the exterior product relation forces any wedge product containing two identical factors to vanish entirely, regardless of how many other factors are present:

Linear Dependence Vanishing

Extending further, a wedge product of vectors v₁,...,vₖ vanishes whenever the set is linearly dependent, since a dependent vector can be written as a combination of the others, and bilinearity reduces this case back to the repeated-factor relation.

The Relation Governing Rank Bound

No Nonzero Product Beyond Dimension

Since any k > n vectors in an n-dimensional space are automatically dependent, the exterior product relation forces:

matching the rank ceiling already established for alternating tensors in general.

Decomposability Relation

Not Every Element Factors as a Simple Wedge

A distinguishing relation at higher degree is that not every element of Λᵏ(V*) can be written as a single wedge product φ₁ ∧ ... ∧ φₖ of covectors; elements that can are called decomposable, and the failure of decomposability for general elements of Λᵏ(V*) at k ≥ 2 when n ≥ 4 is itself a structural fact governed by additional relations (the Plücker relations) beyond the basic anticommutation and associativity rules.

Diagram of the Relation Hierarchy