16.16.3 Tensor Alternating Basis Increasing Index Rule

The Tensor Alternating Basis Increasing Index Rule governs how tensor components change with basis transformations, ensuring antisymmetry in index ordering.

Tensor Alternating Basis Increasing Index Rule is the specific convention requiring that basis wedge products of an exterior power be formed only from index sequences arranged in strictly increasing order, ensuring that each distinct nonzero basis direction of the space is represented exactly once, without redundancy or omission. It is the disambiguation principle that makes the alternating basis construction well-defined and free of duplicate or contradictory representations.

Statement of the Rule

The Strict Ordering Requirement

Given an ordered basis e₁, ..., eₙ of a vector space V, the increasing index rule restricts basis wedge products at degree k to those formed from sequences satisfying:

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

Any wedge product e_{j₁} ∧ ... ∧ e_{jₖ} whose index sequence is not already in this strictly increasing form is excluded from the canonical basis listing, even though it remains a valid, nonzero element of Λᵏ(V) whenever the jᵢ are distinct.

Why Non-Increasing Sequences Are Excluded

Every wedge product formed from a non-increasing but still distinct sequence of indices is expressible as plus or minus a basis element already included under the increasing convention, differing only by the sign of the permutation needed to sort the indices. Including both the sorted and unsorted versions in the basis would introduce a linear dependency, since they differ only by a scalar factor of ±1, violating the requirement that a basis consist of linearly independent elements.

Consequences of the Rule

Elimination of Redundancy

By fixing a single canonical ordering, the increasing index rule guarantees that the resulting set of basis wedge products has no redundant, linearly dependent members, and that its size matches exactly the dimension of Λᵏ(V) as given by the binomial coefficient:

dim (Λ^{k} (V)) = (\binom{n}{k})

Automatic Exclusion of Repeated Indices

A strictly increasing sequence, by its very definition, cannot contain repeated entries, since strict inequality between consecutive terms forbids equality. This means the increasing index rule automatically enforces the repeated factor rule as a side effect: any attempt to list a wedge product with a repeated index is simply not a valid strictly increasing multi-index and is excluded from the basis listing from the outset, rather than needing to be separately identified as vanishing.

Reordering Formula for Non-Canonical Sequences

Any wedge product with distinct but disordered indices can be converted to canonical form using the general reordering identity:

e_{j_{1}} \land \dots \land e_{j_{k}} = sgn (σ) \cdot e_{i_{1}} \land \dots \land e_{i_{k}}

where i₁ < ... < iₖ is the increasing rearrangement of j₁, ..., jₖ, and σ is the sorting permutation, so that every wedge product of distinct basis vectors, in any order, is a signed multiple of exactly one canonical increasing basis element.

Practical Role in Computation

Canonical Coordinate Representation

The increasing index rule provides the fixed reference ordering needed to express any element of Λᵏ(V) uniquely as a coordinate vector relative to the canonical basis, since every coefficient in such an expansion is associated with one, and only one, strictly increasing multi-index.

Consistency in Symbolic and Numerical Software

In symbolic computation and computer algebra systems implementing exterior algebra, the increasing index rule is what allows basis elements to be stored, compared, and combined unambiguously: any wedge product computed internally can always be normalized to canonical increasing form, with an accompanying sign, before being added to or compared against other terms.

Significance of the Rule

The increasing index rule is the disambiguation convention that transforms the abstract collection of all possible wedge products of basis vectors into a well-defined, non-redundant, linearly independent basis for each exterior power. It guarantees the correct dimension count, automatically subsumes the repeated factor rule, and supplies the canonical reference ordering required for consistent coordinate representation and computer implementation of exterior algebra.