16.19.3 Tensor Alternation Repeated Argument Check

Tensor Alternation Repeated Argument Check is the verification step that directly tests whether a candidate multilinear function evaluates to zero whenever two of its argument slots are filled with the same vector, serving as the most fundamental and often the most straightforward test for confirming alternation, distinct from but closely related to the sign-based checks performed by the slot swap and sign change procedures. It targets the vanishing property at the heart of the definition of an alternating tensor.

Statement of the Check

The Vanishing Condition

For a candidate k-linear function T on a vector space V, the repeated argument check verifies that for every choice of vector v and every pair of positions i ≠ j:

with the same vector v placed in positions i and j while the remaining arguments are arbitrary. Passing this check for every such pairing of positions, using a generic vector v, is the direct verification of the defining vanishing property of alternating tensors.

Why Only Two Positions Need Checking at a Time

Although a tensor might have several repeated arguments simultaneously among its k slots, it suffices to check the vanishing condition for exactly one repeated pair at a time. If any two positions share a common vector, the multilinear structure combined with a single vanishing check at those two positions is enough to force the entire expression to zero, regardless of what other arguments are present, since the repeated pair alone triggers the vanishing behavior.

Practical Testing Method

Substituting a Generic Symbolic Vector

In practice, the repeated argument check is carried out by substituting a single symbolic vector, often denoted generically as v, into two chosen argument slots of the candidate function's defining formula, and then simplifying the resulting expression algebraically to confirm it reduces identically to zero, for arbitrary choices of the remaining argument vectors.

Testing Representative Position Pairs

Because the check must hold for every pair of positions, and because multilinear tensors are typically defined by formulas with some structural symmetry across positions, it is often sufficient in practice to verify the check for a single representative pair, such as positions 1 and 2, and argue by the structural symmetry of the defining formula that the same reasoning applies to any other pair.

Connection to the Broader Verification Procedure

Logical Priority Within the Full Procedure

Within the complete alternation verification procedure, the repeated argument check is often performed before or alongside the slot swap check, since the two are equivalent over fields where 2 is invertible, and confirming one directly establishes the other. In such settings, verifying the repeated argument condition can be treated as the primary and often more computationally direct route to certifying alternation.

Necessity in Characteristic Two Settings

As previously established in the broader verification procedure, fields of characteristic two require both checks to be applied independently, since the sign-reversal check does not automatically imply vanishing on repetition in that setting. In characteristic two, the repeated argument check must be treated as the fundamental, non-negotiable defining property, since it is this vanishing condition, not the swap condition, that is used to define alternation in full generality across all fields.

Applying the Check to Known Examples

Verifying the Determinant

Applying the repeated argument check to the determinant function directly confirms a familiar linear algebra fact: a matrix with two identical columns has determinant zero. This is not merely a computational curiosity but the specific instance of the repeated argument check applied to the determinant viewed as an alternating n-linear form on its column vectors.

Verifying a Wedge Product Expression

For an expression built from a wedge product, such as u ∧ v ∧ w, substituting w = u immediately produces u ∧ v ∧ u, and the repeated argument check confirms this evaluates to zero, consistent with the wedge product's inherited alternating structure from the exterior algebra construction.

Significance of the Check

The alternation repeated argument check is the most direct test of the defining vanishing property shared by all alternating tensors, providing a computationally accessible route to certifying alternation that requires only a single symbolic substitution rather than tracking permutation signs. It serves as the foundational check within the broader verification procedure, takes on primary importance in characteristic two settings where sign-based reasoning fails to suffice, and directly confirms familiar facts such as the vanishing of determinants with repeated columns.