4.20.2 Tensor Alternating Repeated Argument Result

Tensor Alternating Repeated Argument Result is the fact that an alternating multilinear map necessarily outputs zero whenever two of its argument slots are filled with the same vector: f(..., v, ..., v, ...) = 0. This result is the most primitive form of the alternating condition, more fundamental than the sign-change rule it is equivalent to outside characteristic 2, and it is the single fact from which the entire theory of alternating maps, exterior powers, and determinants is built.

Statement of the Result

The Core Fact

If f: V × ... × V → W is alternating and the vector v occupies two distinct argument slots i and j simultaneously, with the remaining n - 2 slots filled arbitrarily, then

with v marked at positions i and j. This is taken, in most treatments, as the defining condition of alternation itself, with the sign-change behavior under transposition derived from it as a consequence rather than the reverse.

Why This Is the Primitive Definition

The repeated-argument result is preferred as the basic definition because it remains meaningful and strong in every characteristic, including characteristic 2, where the sign-change condition degenerates to ordinary symmetry and so fails to capture the intended notion; the vanishing condition, by contrast, continues to say something nontrivial regardless of the characteristic of the field.

Immediate Consequences

Vanishing on Linearly Dependent Tuples

If v₁, ..., vₙ are linearly dependent, some vₖ can be written as ∑_{i≠k} cᵢvᵢ. Expanding f(v₁,...,vₙ) by linearity in slot k produces a sum of terms, each of which has vᵢ repeated in two slots (position i and position k) for some i, and each such term vanishes by the repeated-argument result. Hence f(v₁,...,vₙ) = 0 for every linearly dependent tuple, extending the basic result far beyond the case of a literal repeated vector.

Deriving the Sign-Change Rule

Expanding f(...,v+w,...,v+w,...) = 0, obtained by applying the repeated-argument result to the sum v + w placed in both of two slots, by multilinearity produces four terms, two of which vanish (each has v or w repeated), leaving f(...,v,...,w,...) = -f(...,w,...,v,...) in every characteristic other than 2. The sign-change rule is thus a derived consequence, not an independent assumption.

Concrete Instances of the Result

Determinant With a Repeated Row or Column

The determinant of a matrix with two identical rows, or two identical columns, is zero; this is the repeated-argument result applied directly to the determinant viewed as an alternating multilinear function of its rows or of its columns, and it is one of the most frequently invoked facts in elementary linear algebra, often used to justify that a matrix with a repeated row is singular.

The Levi-Civita Symbol

The Levi-Civita symbol ε_{i₁...iₙ}, the component array of the standard alternating n-form on an n-dimensional space relative to a basis, is defined to be zero whenever any two of its indices are equal; this is the repeated-argument result written out explicitly at the level of components, and it underlies the vanishing of terms in any index sum involving ε where a repeated index appears among the alternating slots.

Top Exterior Power Vanishing Beyond the Dimension

For V of dimension d, any alternating multilinear map of arity n > d must vanish identically on every tuple, because any n > d vectors in a d-dimensional space are automatically linearly dependent, and the repeated-argument result, extended to dependent tuples, forces the output to zero in every case; correspondingly, the exterior power ⋀ⁿV is the zero space whenever n > d.

The Result as a Test for Alternation

Sufficient Condition to Verify

To confirm that a candidate multilinear map is alternating, it suffices to verify the repeated-argument result directly, checking that the map vanishes whenever two adjacent slots hold the same vector; the general vanishing on any two coinciding slots, and the full permutation sign rule, then follow automatically without additional verification.

Failure of the Result Rules Out Alternation

If a multilinear map fails to vanish on some tuple with a repeated vector, it cannot be alternating, regardless of whether it happens to satisfy the sign-change rule for some specific pairs of arguments; the repeated-argument result is thus a clean, unambiguous test that either confirms or immediately rules out the alternating property for a given multilinear map.

Structural Role in the Exterior Power Construction

Defining Relation of the Exterior Power

The exterior power ⋀ⁿV is constructed as the quotient of V ⊗ ... ⊗ V by the subspace generated by all elementary tensors v₁ ⊗ ... ⊗ vₙ in which two of the vᵢ coincide; the repeated-argument result is, at the level of the quotient construction, exactly the statement that this subspace is precisely what must be collapsed to zero for the induced wedge product v₁ ∧ ... ∧ vₙ to represent an alternating multilinear map.

Guaranteeing the Universal Property for Alternating Maps

Because the repeated-argument result identifies exactly which elements of V ⊗ ... ⊗ V must vanish for an induced map to be alternating, it is the condition that makes the exterior power the correct universal recipient for alternating multilinear maps, playing the same role for alternating maps that the general multilinearity relations play for the ordinary tensor product's universal property.

Distinguishing the Result From Related But Weaker Conditions

Not the Same as Vanishing on a Single Zero Vector

The repeated-argument result concerns two slots sharing the same nonzero vector, not a slot being filled with the zero vector; multilinearity alone already forces f(...,0,...) = 0 whenever any single slot is zero, a much weaker and separately guaranteed fact that holds for every multilinear map, alternating or not.

Not Automatically True for General Multilinear Maps

A general multilinear map need not vanish when two arguments coincide; the bilinear form f(v,w) = v₁w₂ on R² (using coordinates) is a straightforward multilinear map for which f(v,v) = v₁v₂ is generally nonzero, illustrating that the repeated-argument result is a genuine additional constraint, not an automatic feature of multilinearity itself.