16.7.1 Tensor Repeated Argument Detection

Tensor Repeated Argument Detection is the practical procedure and criterion for determining, given a specific list of vectors fed into an alternating tensor, whether any two of them coincide or are otherwise linearly dependent in a way that guarantees the tensor's output must be zero — turning the abstract vanishing property into a concrete, checkable test.

Direct Equality Detection

The Simplest Test

The most immediate form of detection checks whether any two argument vectors are literally identical:

This detection requires only pairwise comparison of the k input vectors, an O(k²) check in the number of arguments, and needs no knowledge of the tensor's internal structure beyond the fact that it is alternating.

Component-Index Analogue

At the level of index notation, the analogous detection checks whether any two index labels within a component symbol coincide, immediately certifying that component as zero without needing to evaluate anything further:

Linear Dependence Detection

Beyond Literal Equality

Repeated-argument vanishing extends beyond exact equality: if any argument can be written as a linear combination of the others, the tensor also vanishes. Detection in this broader sense requires checking whether the set of input vectors is linearly independent:

Practical Detection via Matrix Rank

Arranging the k vectors as columns of a matrix, detection reduces to computing whether the matrix has full column rank; if the rank is less than k, a linear dependence is detected and the tensor evaluates to zero without needing to invoke the alternating tensor's formula at all:

Detection via the Determinant Instance

Direct Application at Top Rank

For the top-rank alternating tensor (the determinant), repeated-argument detection is exactly the classical test for singularity: a square matrix has determinant zero precisely when its columns fail linear independence, so detecting a repeated or dependent column is equivalent to detecting a vanishing determinant.

Efficient Detection without Full Evaluation

Because detecting linear dependence via row reduction is computationally cheaper than expanding a full determinant formula for large k, repeated-argument detection serves as an efficient shortcut: a dependence check can certify the tensor's value is zero long before any full antisymmetrized sum would need to be computed.

When Detection Fails to Apply

Genuinely Independent Arguments

If no dependence is detected — the k vectors are linearly independent — repeated-argument detection yields no information about the tensor's value one way or another; the tensor may be zero or nonzero depending on its specific definition, and a positive detection result (independence) is not by itself a certificate of nonvanishing.

Detection Is One-Directional

Repeated-argument detection is a sufficient condition for vanishing, not a necessary one: an alternating tensor can still evaluate to zero on a linearly independent set of arguments if the specific tensor happens to have a zero value there, so absence of detected dependence should not be mistaken for a guarantee of a nonzero result.

Diagram of the Detection Procedure