16.5.5 Tensor Alternating Component Reconstruction Rule

Tensor Alternating Component Reconstruction Rule is the explicit formula for recovering the value of any component of an alternating tensor, given only the small set of independent values stored for canonically ordered index tuples, closing the loop opened by redundancy reduction by specifying exactly how the discarded information is regenerated on demand.

The Reconstruction Formula

From Canonical Value to Arbitrary Component

Given the independent value T_{j₁<j₂<...<jₖ} stored for the increasing index tuple (j₁ < j₂ < ... < jₖ), the component for any permutation σ of that same tuple is reconstructed by:

and for any index tuple containing a repeated label, the reconstruction rule simply returns zero without consulting the stored table at all.

The Rule as a Two-Step Lookup Procedure

Reconstructing an arbitrary component T_{i₁...iₖ} proceeds in two steps: first check whether any two of i₁,...,iₖ coincide, returning 0 immediately if so; otherwise, sort the tuple into increasing order, recording the permutation σ used, look up the stored value for the sorted tuple, and multiply by sgn(σ).

Justifying the Rule

Why the Rule Is Forced, Not Chosen

The reconstruction rule is not an independent convention layered on top of the alternating tensor's definition; it is the unique formula consistent with the sign change constraint and the vanishing-on-repetition fact already established for alternating tensors. Any other reconstruction formula would either contradict the sign change constraint for some permutation or fail to reduce to the stored value on the identity permutation.

Well-Definedness Across Decompositions

Because sgn(σ) depends only on the parity of σ and not on which specific sequence of transpositions is used to compute it, the reconstruction rule gives the same answer no matter how the sorting from i₁,...,iₖ to j₁ < ... < jₖ is carried out.

Worked Reconstruction Example

Rank 3 Case

Suppose the independent value T_{124} = 7 is stored for n ≥ 4. To reconstruct T_{412}, first note all indices are distinct, then find the sorting permutation carrying (4,1,2) to (1,2,4): this is a 3-cycle, an even permutation, so sgn(σ) = +1:

By contrast, T_{142} requires the sorting permutation carrying (1,4,2) to (1,2,4), a single transposition of the last two positions, giving sgn(σ) = −1:

Reconstruction in Full Tensor Form

Rebuilding the Levi-Civita Expansion

At top rank k = n, the reconstruction rule applied to the single stored scalar c = T_{1,2,...,n} regenerates the entire component array as T_{i₁...iₙ} = c · ε_{i₁...iₙ}, showing that the Levi-Civita symbol is precisely the reconstruction rule's sign-assignment mechanism made explicit as a standalone array.

Reconstruction and the Antisymmetrization Operator

The reconstruction rule can also be phrased using the antisymmetrization operator applied to an arbitrary seed assignment: assigning the stored values to the canonical tuples and zero elsewhere, then applying Alt(·), regenerates exactly the same component array as the direct sign-and-lookup procedure.

Computational Role of the Rule

Storage-to-Access Tradeoff

The reconstruction rule is what makes the compact C(n,k)-sized storage format practical: rather than materializing the full n^k array, software or manual computation can store only the canonical values and apply the reconstruction rule on demand whenever any arbitrary component is needed, trading a small amount of per-access computation (sorting and sign tracking) for a large reduction in storage.

Diagram of the Reconstruction Pipeline