16.13.1 Tensor Exterior Power Repeated Factor

The tensor exterior power with repeated factor examines how repeated vectors interact, shaping antisymmetric structures in exterior algebra.

Tensor Exterior Power Repeated Factor is the phenomenon by which any wedge product containing two or more identical vector factors, whether appearing directly adjacent or separated within the expression, necessarily evaluates to zero within the exterior power of a vector space. It is the direct algebraic consequence of the alternating property that defines exterior powers, and it serves as the fundamental mechanism that restricts nonzero elements of an exterior power to combinations of linearly independent vectors.

The Repeated Factor Rule

Basic Statement

For a vector space V and any vector v in V, the exterior power construction enforces:

v \land v = 0

This is not an additional axiom imposed on top of multilinearity; it is the defining relation used to construct the exterior algebra as a quotient of the tensor algebra, and every other alternating property follows from it.

Extension to Longer Products

The rule extends immediately to wedge products of any length. If a vector v appears twice among the factors of a k-fold wedge product, regardless of position, the entire product vanishes:

v_{1} \land v \land v_{2} \land \dots \land v \land \dots = 0

This holds because the vectors can always be permuted, at the cost of sign changes, until the two copies of v become adjacent, at which point the adjacent-repetition rule applies directly.

Derivation From Antisymmetry

Sign-Flip Argument

Because the wedge product is antisymmetric, swapping two adjacent factors introduces a sign change:

u \land w = - (w \land u)

Setting w equal to u in this identity gives u ∧ u = −(u ∧ u), which forces 2(u ∧ u) = 0. Over any field where 2 is invertible, this yields u ∧ u = 0 as an immediate consequence, tying the repeated factor rule directly to the antisymmetry axiom.

Characteristic Two Subtlety

In fields of characteristic two, the argument above does not directly force u ∧ u = 0, since 2 equals 0 and the equation 2(u ∧ u) = 0 is automatically satisfied without constraining u ∧ u. For this reason, the exterior algebra in characteristic two is defined by explicitly imposing v ∧ v = 0 as a primary relation rather than deriving it from antisymmetry, which is why the repeated factor rule is treated as the foundational definition rather than a corollary in the general construction of exterior algebras.

Consequences for Linear Dependence

Detecting Dependent Sets

The repeated factor rule generalizes to detect any linearly dependent collection of vectors, not only literal repetitions. If vectors v₁, v₂, ..., vₖ are linearly dependent, meaning one can be written as a combination of the others, then:

v_{1} \land v_{2} \land \dots \land v_{k} = 0

This follows by substituting the dependent vector as a linear combination of the others, distributing the wedge product across the sum by multilinearity, and observing that every resulting term contains a repeated factor and therefore vanishes.

Nonzero Products as Independence Certificates

Conversely, whenever a wedge product of k vectors is nonzero, those vectors must be linearly independent. This gives the repeated factor rule its practical role as a computational test: evaluating a wedge product provides a direct algebraic certificate of whether a given set of vectors spans a genuine k-dimensional subspace.

Role in Bounding Exterior Power Dimension

Truncation Beyond the Ambient Dimension

Since any collection of more than n vectors in an n-dimensional space must be linearly dependent, the repeated factor rule guarantees that every wedge product of more than n vectors is automatically zero. This is the mechanism responsible for the vanishing of Λᵏ(V) whenever k exceeds n, linking the repeated factor property directly to the dimension formula governing exterior powers.

Summary of Its Structural Role

The repeated factor property is the single relation from which the entire alternating character of exterior powers derives. It explains why identical or dependent vectors cannot contribute nonzero volume elements, it underlies the antisymmetric sign behavior of the wedge product, and it directly produces the dimension bound that limits nonzero exterior powers to grades no higher than the dimension of the underlying vector space.