16.5.3 Tensor Alternating Component Repeated Index Vanishing

Tensor Alternating Component Repeated Index Vanishing occurs when an alternating tensor's component becomes zero if indices repeat, enforcing antisymmetry in algebra.

Tensor Alternating Component Repeated Index Vanishing is the specific phenomenon by which any component of an alternating tensor automatically equals zero whenever two or more of its index labels take the same value, isolated here as a standalone fact with its own derivation, its own consequences for storage and computation, and its own edge cases at the boundary of characteristic-2 fields.

The Vanishing Statement

Precise Formulation

For an alternating tensor T of rank k, any component in which two index positions carry an identical label vanishes:

T_{\dots i \dots i \dots} = 0

regardless of which two positions carry the repeated value i, and regardless of what values the remaining indices take.

Derivation from the Sign Change Constraint

The vanishing follows immediately from swapping the two repeated-label positions: since swapping identical labels changes nothing about the symbol, but the sign change constraint still demands a factor of −1:

T_{\dots i \dots i \dots} = - T_{\dots i \dots i \dots} \Rightarrow 2 T_{\dots i \dots i \dots} = 0

so the value must be zero whenever 2 is invertible in the underlying field.

Consequences for Which Slots Can Be Nonzero

Nonzero Requires All-Distinct Indices

The vanishing fact means the only components with any chance of being nonzero are those where all k index labels are pairwise distinct; every other slot in the raw n^k array is automatically zero without needing to be individually computed or stored.

Fraction of Slots Eliminated

For rank k in dimension n, the fraction of the n^k raw slots eliminated by repeated-index vanishing grows quickly with k; already at k = 2, exactly n out of n² slots (the diagonal) are eliminated, and at higher rank the elimination fraction approaches 1 as k approaches n, since fewer and fewer index assignments avoid any repetition.

fraction with a repeat = 1 - \frac{n! / (n - k)!}{n^{k}}

Rank Bound as a Direct Consequence

No Nonzero Tensor of Rank Above the Dimension

If k > n, then any assignment of k index values drawn from only n available labels must, by the pigeonhole principle, repeat at least one label. Combined with the repeated-index vanishing fact, this forces every single component of a rank-k > n alternating tensor to be zero:

k > n \Rightarrow Λ^{k} (V) = {0}

This is arguably the single most important consequence of repeated-index vanishing, since it caps the entire exterior algebra at a finite number of nonzero graded pieces.

The Characteristic-2 Exception

Where the Derivation Breaks Down

The derivation 2T = 0 ⇒ T = 0 requires that 2 be invertible. In characteristic 2, this step fails, and repeated-index vanishing is no longer automatic from the antisymmetric sign relation alone; it must instead be imposed as a separate condition if the alternating (rather than merely antisymmetric) property is genuinely intended.

Practical Irrelevance in Standard Settings

Because virtually all applications in geometry, physics, and standard linear algebra work over characteristic-0 fields, this exception is noted for completeness but does not affect the vanishing conclusion in the settings where alternating tensors are ordinarily used.

Diagram of Vanishing Slots

The filled black squares mark the positions where index repetition forces vanishing; only the off-diagonal, all-distinct-index cells carry potentially nonzero values.