16.7.4 Tensor Repeated Argument Component Vanishing

Tensor Repeated Argument Component Vanishing is the translation of the abstract zero repeated argument rule into the language of indexed components, obtained by substituting standard basis vectors into the tensor's argument slots and reading off exactly which component symbols the rule forces to zero.

From Abstract Vectors to Basis Substitution

Setting Up the Substitution

Given a basis e₁, ..., eₙ for V, any component of an alternating tensor T is obtained by evaluating T directly on a chosen tuple of basis vectors:

The zero repeated argument rule applies directly to this substitution whenever two of the chosen basis vectors coincide.

Direct Substitution of the Rule

If any two index labels iₐ and iᵦ are equal, the corresponding basis vectors e_{iₐ} and e_{iᵦ} fed into T are literally the same vector, and the zero repeated argument rule applies immediately:

so component vanishing is not a new fact requiring separate proof; it is a direct readout of the abstract rule under this particular substitution.

Why the Translation Is Exact, Not Approximate

No Information Lost in Substitution

Because a basis is, by definition, a set of linearly independent vectors spanning V, substituting basis vectors into the tensor's arguments captures the tensor's behavior completely: the entire component array, together with multilinearity, determines T on all of V, not just on basis vectors. This means the component vanishing derived here is fully equivalent to, not merely suggestive of, the abstract rule.

Basis-Independence of the Vanishing Conclusion

Although the specific labels i₁,...,iₖ that trigger vanishing depend on which basis is chosen, the fact that some components vanish under repetition holds for every basis, since the zero repeated argument rule itself makes no reference to any particular basis; changing basis simply relabels which components are which; it never removes the vanishing pattern.

Consequences Specific to the Component Picture

Immediate Identification of Zero Entries

Once a basis is fixed, the component vanishing translation gives an immediate, purely combinatorial way to identify every zero entry in the component array without evaluating T at all: any index tuple with a repeated label names a zero component automatically.

Reduction to Distinct-Index Storage

Combined with the sign change component effect for distinct-index components, this vanishing translation is the reason a compact storage scheme needs only track values for tuples of distinct, increasing indices — every other tuple is either a sign-flipped duplicate or an automatic zero from repetition.

Worked Example

Rank 2, Dimension 3

For a rank-2 alternating tensor on a 3-dimensional space with basis e₁, e₂, e₃, substituting i = j = 2 gives:

directly from the rule, with no need to invoke the sign-reversal law or any other derived property; the substitution alone suffices.

Diagram of the Translation Path