16.3.4 Tensor Alternating Type Requirement

Tensor Alternating Type Requirement is the set of formal conditions a multilinear map must satisfy before it can be classified as an alternating tensor, fixing the exact test that separates alternating tensors from the larger family of general multilinear maps and from the narrower family of symmetric tensors.

Baseline Requirement: Multilinearity

Linearity in Each Argument Separately

Before any alternation condition is checked, the object must already satisfy the ordinary type requirement of a multilinear map: linear in each slot when all other slots are held fixed.

Without this baseline, the alternation condition below is not even well formed, since permutation-sign rules presuppose a linear structure in every slot.

The Alternation Condition Itself

Sign Reversal Under Transposition

The defining type requirement is that swapping any two adjacent arguments negates the value:

Equivalent Vanishing-on-Repetition Form

In characteristics other than 2, this requirement is exactly equivalent to demanding the tensor vanish whenever two of its arguments are equal:

Both forms are accepted as satisfying the type requirement, and each can be derived from the other, so the requirement is stated in whichever form is more convenient for the object under test.

Full Permutation Consequence

Because every permutation decomposes into transpositions, satisfying the adjacent-swap condition forces the general permutation law to hold for any σ in the symmetric group S_k:

so verifying the type requirement only for adjacent transpositions is sufficient to certify it for all permutations.

Type Requirement by Degree

Degree 0 and Degree 1

At degree 0, the requirement is vacuous, since there are no argument pairs to swap; every scalar trivially qualifies. At degree 1, the requirement is again vacuous for the same structural reason, so every covector is automatically alternating.

Degree 2 and Above

The requirement becomes substantive starting at degree 2, where it excludes any bilinear form that is not skew-symmetric:

A bilinear form satisfying only this equation but not the stronger symmetric one is admitted; one satisfying both simultaneously is forced to be identically zero whenever the field characteristic is not 2, since T(u,v) = T(v,u) = −T(u,v) implies 2T(u,v) = 0.

Degree Above Dimension

A further consequence of the type requirement combined with linear dependence is that no nonzero alternating tensor of degree greater than n = dim(V) can exist, because any k > n arguments drawn from an n-dimensional space must include a repeated direction, forcing the value to zero by the vanishing-on-repetition form.

Distinguishing the Requirement from Related Ones

Contrast with the Symmetric Requirement

The symmetric type requirement demands the opposite sign behavior:

so the alternating and symmetric type requirements are mutually exclusive for any nonzero tensor of degree 2 or higher outside characteristic 2, meaning a tensor satisfying one requirement generically fails the other.

Contrast with No Symmetry Requirement

A general multilinear map imposes no relation at all between T(..., v_i, v_{i+1}, ...) and T(..., v_{i+1}, v_i, ...); the alternating type requirement is therefore a strict subtype, and every alternating tensor is also a general tensor, but not conversely.

Verification Diagram