16.12.1 Tensor Wedge Product Factor Selection

Tensor Wedge Product Factor Selection involves choosing factors that define the antisymmetric structure of tensor products in algebraic contexts.

Tensor Wedge Product Factor Selection is the combinatorial process of choosing which basis covectors, and in what index order, are wedged together to build the standard basis elements of Λᵏ(V*), fixing the convention that determines exactly one representative ordering per subset and thereby avoiding the redundancy that would otherwise arise from listing every possible factor arrangement.

The Selection Problem

Too Many Orderings Without a Convention

Given n basis covectors e₁*, ..., eₙ*, forming a degree-k basis wedge requires selecting k of them and wedging them together; naively, there are n!/(n−k)! ordered ways to select and arrange k factors from n, but the graded anticommutation relation makes many of these orderings redundant — differing only by a sign already determined by the reordering permutation.

The Selection Convention: Increasing Index Order

The standard resolution is to select factors and always list them in strictly increasing index order:

e_{i_{1}}^{*} \land \dots \land e_{i_{k}}^{*}, i_{1} < i_{2} < \dots < i_{k}

This selection convention picks exactly one representative from each equivalence class of orderings related by permutation, since every other ordering of the same subset can be recovered from this one by multiplying by the appropriate sign.

Counting the Selections

The Binomial Count of Valid Selections

The number of distinct factor selections satisfying the increasing-order convention is exactly the number of k-element subsets of an n-element index set:

# selections = (\binom{n}{k})

matching exactly the dimension of Λᵏ(V*), confirming that the selection convention produces precisely one basis element per dimension, with no gaps and no redundancy.

Selection as a Bijection with Basis Elements

The map from k-element subsets of {1,...,n} to basis wedge elements, sending each subset to its increasing-order wedge product, is a bijection onto a basis of Λᵏ(V*); this bijection is the combinatorial heart of the factor selection process.

Worked Selection Example

Selecting Factors for Degree 2 in Four Dimensions

For n = 4, k = 2, the valid selections (subsets of size 2 from {1,2,3,4}) are {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, giving exactly C(4,2) = 6 basis bivectors:

e_{1}^{*} \land e_{2}^{*}, e_{1}^{*} \land e_{3}^{*}, \dots, e_{3}^{*} \land e_{4}^{*}

Non-Selected Orderings Are Not Lost

Recoverable via the Sign Rule

Selections not following the increasing-order convention, such as e₂*∧e₁*, are not additional basis elements; they are recoverable from the selected ones via the sign rule, e₂*∧e₁* = −e₁*∧e₂*, so no information is lost by restricting attention only to increasing-order selections — every possible wedge of two distinct basis covectors is a scalar multiple of exactly one selected basis element.

Selection Does Not Apply to Repeated Factors

Factor selection as described here only concerns subsets of distinct indices; any attempt to select a repeated index (such as e₁*∧e₁*) produces zero directly by the fundamental vanishing relation, so repeated selections are excluded from the counting and basis-construction process entirely rather than contributing degenerate basis elements.