16.4.4 Tensor Antisymmetric Alternating Equivalence Context

Tensor Antisymmetric Alternating Equivalence Context is the specific set of preconditions — multilinearity, field characteristic, and finite rank — under which the antisymmetric condition and the alternating condition on a tensor can be formally proven to imply one another, establishing the precise logical bridge between the two formulations rather than merely asserting they coincide.

Precondition: A Multilinear Setting

Why Multilinearity Must Be Assumed First

Both directions of the equivalence proof manipulate T by substituting sums of vectors into its arguments and expanding using linearity. Without multilinearity already established, expressions like T(..., u + v, ..., u + v, ...) cannot be expanded into a sum of four separate terms, and the entire equivalence argument fails to get started.

This bilinear expansion, applied to the two slots shown, is the engine that drives the equivalence in both directions.

Direction One: Alternating Implies Antisymmetric

The Full Derivation

Assume T(..., w, ..., w, ...) = 0 for all w. Substituting w = u + v into the two matching slots and expanding via the multilinear precondition:

The first and last terms vanish by the alternating hypothesis applied directly to u and to v, leaving:

which rearranges to the antisymmetric condition, with no additional hypothesis on field characteristic required.

Direction Two: Antisymmetric Implies Alternating

Where the Characteristic Enters

Assume T(..., u, ..., v, ...) = −T(..., v, ..., u, ...). Setting u = v = w:

giving 2·T(...,w,...,w,...) = 0. This is only sufficient to conclude T(...,w,...,w,...) = 0 when 2 is invertible in the underlying field — i.e., the field's characteristic is not 2. This is the exact point where the equivalence context becomes conditional rather than unconditional.

The Context Where the Second Direction Fails

In a field of characteristic 2, 2·T = 0 holds automatically regardless of the value of T, so no conclusion about T(...,w,...,w,...) can be drawn; the antisymmetric hypothesis alone becomes too weak to force alternation, and a genuinely antisymmetric-but-not-alternating tensor can exist.

Summary Context Table

The asymmetry between the two directions of implication — one unconditional, one conditional on characteristic — is the central content of this equivalence context, and it is easy to overlook if the two conditions are casually assumed interchangeable without checking which direction of the proof is being invoked.

Rank Context for the Equivalence

The Equivalence Requires at Least Two Slots

Both directions of the proof require at least two argument slots to substitute u and v into; for rank 0 or rank 1 tensors, neither condition is well posed as a nontrivial statement, so the equivalence context is vacuously satisfied rather than actively established by the derivation above.

Diagram of the Equivalence Context