16.13.3 Tensor Exterior Power Basis Construction

The tensor exterior power basis construction provides a structured framework for building bases in exterior algebras through wedge products and antisymmetrization.

Tensor Exterior Power Basis Construction is the systematic procedure by which a basis for the k-th exterior power of a vector space is built directly from a basis of the original space, using strictly increasing sequences of basis indices combined through the wedge product. It provides the explicit combinatorial recipe that turns the abstract existence of exterior powers into a concrete, computable vector space with a well-defined basis and coordinate system.

Starting Data

The Underlying Basis

Let V be an n-dimensional vector space with an ordered basis e₁, e₂, ..., eₙ. The basis construction for Λᵏ(V) begins entirely from this data, since every element of the exterior power can ultimately be expressed as a linear combination of wedge products formed from these basis vectors.

Choosing Index Sequences

The construction proceeds by selecting all strictly increasing sequences of k indices drawn from the set {1, 2, ..., n}:

i_{1} < i_{2} < \dots < i_{k}

Each such sequence determines one basis element of Λᵏ(V), formed as the wedge product of the corresponding basis vectors.

The Resulting Basis Elements

Basis Vectors of the Exterior Power

For each strictly increasing sequence, the associated basis element is:

e_{i_{1}} \land e_{i_{2}} \land \dots \land e_{i_{k}}

The collection of all such elements, ranging over every strictly increasing index sequence of length k, forms a basis of Λᵏ(V). This means every k-vector in the space can be written uniquely as a linear combination of these basis wedge products.

Why Strict Increase Suffices

Any wedge product of basis vectors that is not already in strictly increasing order can be rearranged into strictly increasing order using the antisymmetry property, at the cost of a sign determined by the permutation required. Any wedge product with a repeated index vanishes entirely by the repeated factor rule. Consequently, restricting attention to strictly increasing sequences captures every distinct basis direction exactly once, without redundancy or omission.

Coordinates in the Exterior Power Basis

General Element Expansion

An arbitrary element ω of Λᵏ(V) is expressed in this basis as:

ω = \sum_{i_{1} < \dots < i_{k}} a_{i_{1} \dots i_{k}} e_{i_{1}} \land \dots \land e_{i_{k}}

where the coefficients a with multi-index subscripts are scalars uniquely determined by ω, since the basis wedge products are linearly independent.

Counting the Basis

The number of basis elements produced by this construction equals the number of strictly increasing index sequences of length k drawn from n indices, which is precisely the binomial coefficient C(n, k). This confirms directly, through an explicit basis rather than an abstract argument, the dimension relation governing exterior powers.

Worked Example in Low Dimension

Case n = 3, k = 2

Given basis vectors e₁, e₂, e₃, the strictly increasing pairs of indices are (1,2), (1,3), and (2,3), yielding the basis of Λ²(V):

e_{1} \land e_{2}, e_{1} \land e_{3}, e_{2} \land e_{3}

This matches the predicted dimension C(3, 2) = 3, and any other wedge product of two basis vectors, such as e₂ ∧ e₁, is expressible in terms of this basis via the sign relation e₂ ∧ e₁ = −(e₁ ∧ e₂).

Significance of the Construction

The basis construction transforms exterior powers from an abstract quotient of the tensor algebra into a concrete finite-dimensional space with explicit coordinates. It confirms the binomial dimension formula constructively, provides the practical means to represent k-vectors and differential forms numerically, and establishes the ordering convention that underlies determinant expansions and orientation choices throughout multilinear algebra.