4.20.4 Tensor Alternating Basis Component Pattern

The Tensor Alternating Basis Component Pattern describes how alternating tensors decompose into basis components, revealing structure in multilinear algebra.

Tensor Alternating Basis Component Pattern is the specific structure exhibited by the coordinate array of an alternating multilinear map once a basis is chosen: the array is completely antisymmetric under permutation of its indices, and consequently is determined entirely by its values on strictly increasing tuples of indices, with every other entry obtained from these by the sign of the reordering permutation.

The Component Array of an Alternating Map

Full Antisymmetry

For an alternating multilinear map f: V × ... × V → W and a basis {e₁, ..., e_d} of V, the component array

T_{i_{1} \dots i_{n}} = f (e_{i_{1}}, \dots, e_{i_{n}})

satisfies

T_{i_{σ (1)} \dots i_{σ (n)}} = sgn (σ) T_{i_{1} \dots i_{n}}

for every permutation σ, and T_{i₁...iₙ} = 0 whenever any two indices coincide, both direct restatements, at the level of the array, of the alternating pattern already present in f itself.

Zero Entries Whenever Indices Repeat

The vanishing part of the pattern means the array has nonzero entries only at positions where all n indices i₁, ..., iₙ are pairwise distinct; every entry with a repeated index is automatically zero, regardless of the map f, purely as a consequence of alternation.

Reduction to Strictly Increasing Index Tuples

One Representative Per Set of Indices

Since every entry is determined, up to sign, by the entry with the same set of indices arranged in increasing order, the entire component array is determined by its values on strictly increasing tuples i₁ < i₂ < ... < iₙ. Any other ordering of the same index set is recovered by multiplying by the sign of the permutation needed to sort it into increasing order.

Counting Independent Entries

The number of strictly increasing tuples i₁ < ... < iₙ drawn from {1, ..., d} is the binomial coefficient C(d, n), so an alternating n-linear map on a d-dimensional space is determined by exactly C(d, n) independent scalars, in sharp contrast to the dⁿ entries needed for a general multilinear map before any symmetry is imposed.

Basis of the Exterior Power From This Pattern

Wedge Products of Increasing Index Tuples

The reduction to strictly increasing tuples matches exactly the standard basis of the exterior power ⋀ⁿV, given by the wedge products e_{i₁} ∧ ... ∧ e_{iₙ} for i₁ < ... < iₙ; these C(d,n) elements form a basis precisely because any other wedge product of basis vectors, with indices out of order or repeated, reduces to a sign multiple of one of these or to zero.

Matching Dimension Counts

The dimension of ⋀ⁿV, equal to C(d,n), agrees exactly with the number of independent entries in the component array of an alternating n-linear map on V, confirming from two directions, the array's antisymmetry pattern and the basis count of the exterior power, that alternating multilinear maps of arity n on a d-dimensional space correspond to a space of dimension C(d,n).

The Levi-Civita Symbol as the Extremal Case

Full Antisymmetric Symbol at Top Degree

When n = d, there is, up to scalar multiple, only one nonzero alternating d-linear form on a d-dimensional space, since C(d,d) = 1; its component array is the Levi-Civita symbol ε_{i₁...i_d}, equal to +1 for even permutations of (1,...,d), -1 for odd permutations, and 0 whenever any index repeats, giving the extremal, most constrained instance of the alternating basis component pattern.

Determinant Expressed via the Symbol

The determinant of a d × d matrix M is expressed using this fully antisymmetric symbol as

det (M) = \sum i_{1}, \dots, i_{d} ε_{i_{1} \dots i_{d}} M_{1 i_{1}} \dots M_{d i_{d}}

directly instantiating the general alternating basis component pattern in the single most important special case.

Practical Uses of the Pattern

Storing Only the Independent Entries

Because entries outside strictly increasing index order are redundant, an alternating multilinear map on a computer can be stored using only its C(d,n) independent values rather than the full dⁿ-entry array required for a general multilinear map, giving a substantial reduction in storage that grows more pronounced as n approaches d.

Reconstructing Any Entry From the Stored Ones

Given the stored values on increasing tuples, any entry T_{i₁...iₙ} with a possibly disordered but non-repeating index tuple is reconstructed by sorting the indices into increasing order, counting the parity of the sorting permutation, and multiplying the corresponding stored value by that sign; any tuple with a repeated index is known immediately to be zero without further lookup.

Verifying a Component Array Represents an Alternating Map

A proposed component array T_{i₁...iₙ} is confirmed to represent an alternating multilinear map exactly when it satisfies the full antisymmetry condition under permutation and vanishes on repeated indices; checking this reduces to comparing entries related by a single transposition, since the general permutation case follows once the transposition case is established.